Inertial Measurement Units (IMUs) enable portable, multibody motion capture in diverse environments beyond the laboratory, making them a desirable choice for diagnosing mobility disorders and supporting rehabilitation in clinical or home settings. However, challenges associated with IMU measurements, including magnetic distortions and errors due to integration drift, complicate their broader use for motion capture. In this work, we propose a tightly coupled motion-capture approach that directly integrates IMU measurements with multibody dynamic models via an iterated extended Kalman filter to simultaneously estimate the system’s kinematics and kinetics. By enforcing the complete multibody system dynamics and utilizing only accelerometer and gyroscope data, our method accurately estimates joint kinematics and kinetics. Our algorithm is designed to fuse different sensor data, such as optical motion-capture measurements and joint torque readings, to further enhance estimation accuracy. We validated our approach using highly accurate ground-truth data from a 3-degree-of-freedom pendulum and a 6-degree-of-freedom collaborative robot. We demonstrate a maximum root-mean-square difference of 3.75° in the pendulum’s computed joint angles with respect to the marker motion-capture inverse kinematics. For the robot, we observed a maximum joint angle root-mean-square difference of 3.24° with respect to the joint encoders, while the maximum joint angle root-mean-square difference of the optical motion-capture inverse kinematics with respect to the encoders was 1.16°. With regard to kinetic estimates, we report a maximum joint torque root-mean-square difference of 3.02 Nm in the pendulum with respect to the marker motion-capture inverse dynamics and 4.27 Nm in the robot relative to its joint torque sensors.

Inertial measurement units (IMUs) are a promising means to capture human motion, yet obtaining meaningful biomechanical quantities from IMU measurements remains non-trivial. This tutorial-style review focuses on kinematics and introduces four key aspects (inertial human motion capture objective, environmental conditions, subject & attributes, and motion characteristics) to determine how to translate biomechanical problems into adequate formulations for the fusion of inertial sensor measurements. We identify three fundamental challenges for kinematics estimation from IMUs: IMUs do not provide direct information about the joint angle, IMUs do not measure their own orientation, and real-world environments and dynamics compromise sensor reliability. Though there exist widely-used methods to overcome these challenges, they suffer from severe limitations in real-life applications, e.g., the need for sensor-to-segment calibration, and the fact that magnetic field disturbances degrade joint angle accuracy. The full potential for many use-cases hence remains untapped in terms of accuracy and reliability. We share insights into recently proposed methods, e.g., exploiting the human body's kinematic chain constraints, having the potential to overcome these limitations. We also present guiding questions related to the four key aspects and illustrate their use for navigating the methodological landscape for the use-case of lower-extremity joint angle estimation, for which we share open-access code and compare the traditional workflow with three alternatives. Our aim is to bridge the gap between the sensor fusion community developing methods for human motion capture and the biomechanics community in need of accurate, easy-to-use, and reliable methods to study human motion outside of the laboratory.

This article addresses sequential Bayesian filtering for nonlinear and stochastic dynamical systems. We extend a Galerkin-approach that was previously used for the prediction of non-Gaussian probability density functions, to incorporate linear and non-linear measurement updates. The proposed method results in a linear pipeline of prediction and update steps, which are computed as sparse matrix operations on the finite-dimensional coefficient vector. The performance of our approach is demonstrated in numerical experiments for nonlinear dynamical 2D- and 4D-systems, using results of a standard particle filter as reference, both in terms of accuracy and computational expenses.

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.

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

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.