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.

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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.

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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.

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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.

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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.@en

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.

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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.

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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.

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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.

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Trunk motor control is essential for the proper functioning of the upper extremities and is an important predictor of gait capacity in children with delayed development. Early diagnosis and intervention could increase the trunk motor capabilities in later life, but current tools used to assess the level of trunk motor control are largely subjective and many lack the sensitivity to accurately monitor development and the effects of therapy. Inertial measurement units could yield an objective quantitative assessment that is inexpensive and easy-to-implement. We hypothesized that root mean square of jerk, a proxy for movement smoothness, could be used to distinguish age and thereby presumed motor development. We attached a sensor to the trunks of six young children with no known developmental deficits. Root mean square of jerk decreases with age, up to 24 months, and is correlated to a more established method, i.e., center-of-pressure velocity, as well as other standard inertial measurement unit outputs. This metric therefore shows potential as a method to differentiate trunk motor control levels.

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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.@en

The last years have seen a growing body of literature on data-driven pedestrian inertial navigation. However, despite this, it is still unclear how to efficiently combine classical models and other a priori information with existing machine learning frameworks. In this paper, we first categorize existing approaches to data-driven pedestrian inertial navigation, including approaches where a machine learning algorithm is embedded into an overarching classical framework and purely data-driven frameworks. We then propose an estimation framework where navigation estimates obtained by classical means are fed to a machine learning algorithm which is trained to correct and improve the estimates. Further, we describe three symmetries that can be used to constrain the proposed estimation framework and thereby improve its performance. These are 1) the rotational symmetry of pedestrian dynamics, 2) the rotational symmetry of the sensors, and 3) the temporal symmetry of pedestrian dynamics. To demonstrate the usefulness of the proposed framework, we use data from foot-mounted inertial sensors utilizing zero-velocity updates under mixed walking and running. Machine learning corrections are implemented using both neural networks and Gaussian processes.

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In this paper, a simultaneous localization and mapping algorithm for tracking the motion of a pedestrian with a foot-mounted inertial measurement unit is proposed. The algorithm uses two maps, namely, a motion map and a magnetic field map. The motion map captures typical motion patterns of pedestrians in buildings that are constrained by e.g. corridors and doors. The magnetic map models local magnetic field anomalies in the environment using a Gaussian process model and uses them as position information. These maps are used in a Rao-Blackwellized particle filter to correct the pedestrian position and orientation estimates from the pedestrian dead-reckoning. The pedestrian dead-reckoning is computed using an extended Kalman filter with zero-velocity updates. The algorithm is validated using experimental sequences and the results show the efficacy of the algorithm in localizing pedestrians in indoor environments.@en

We present a computationally efficient algorithm for using variations in the ambient magnetic field to compensate for position drift in integrated odometry measurements (dead-reckoning estimates) through simultaneous localization and mapping (SLAM). When the magnetic field map is represented with a reduced-rank Gaussian process (GP) using Laplace basis functions defined in a cubical domain, analytic expressions of the gradient of the learned magnetic field become available. An existing approach for magnetic field SLAM with reduced-rank GP regression uses a Rao-Blackwellized particle filter (RBPF). For each incoming measurement, training of the magnetic field map using an RBPF has a computational complexity per time step of O(NpN2m), where Np is the number of particles, and Nm is the number of basis functions used to approximate the Gaussian process. Contrary to the existing particle filter-based approach, we propose applying an extended Kalman filter based on the gradients of our learned magnetic field map for simultaneous localization and mapping. Our proposed algorithm only requires training a single map. It, therefore, has a computational complexity at each time step of O(N2m). We demonstrate the workings of the extended Kalman filter for magnetic field SLAM on an open-source data set from a foot-mounted sensor and magnetic field measurements collected onboard a model ship in an indoor pool. We observe that the drift compensating abilities of our algorithm are comparable to what has previously been demonstrated for magnetic field SLAM with an RBPF. @en

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.

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Real-time motion tracking of kinematic chains is a key prerequisite in the control of, e.g., robotic actuators and autonomous vehicles and also has numerous biomechanical applications. In recent years, it has been shown that, by placing inertial sensors on segments that are connected by rotational joints, the motion of that kinematic chain can be tracked accurately. These methods specifically avoid using magnetometer measurements, which are known to be unreliable since the magnetic field at the different sensor locations is typically different. They rely on the assumption that the motion of the kinematic chain is sufficiently rich to assure observability of the relative pose. However, a formal investigation of this crucial requirement has not yet been presented, and no specific conditions for observability have so far been given. In this work, we present an observability analysis and show that the relative pose of the body segments is indeed observable under a very mild condition on the motion. We support our results by simulation studies, in which we employ a state estimator that neither uses magnetometer measurements nor additional sensors and does not impose assumptions on the accelerometer to measure only the direction of gravity, nor on the range of motion or degrees of freedom of the joints. We investigate the effect of the amount of excitation and of stationary periods in the data on the accuracy of the estimates. We then use experimental data from two mechanical joints as well as from a human gait experiment to validate the observability criterion in practice and to show that small excitation levels are sufficient for obtaining accurate estimates even in the presence of time periods during which the motion is not observable.

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Skin-attached inertial sensors are increasingly used for kinematic analysis. However, their ability to measure outside-lab can only be exploited after correctly aligning the sensor axes with the underlying anatomical axes. Emerging model-based inertial-sensor-to-bone alignment methods relate inertial measurements with a model of the joint to overcome calibration movements and sensor placement assumptions. It is unclear how good such alignment methods can identify the anatomical axes. Any misalignment results in kinematic cross-talk errors, which makes model validation and the interpretation of the resulting kinematics measurements challenging. This study provides an anatomically correct ground-truth reference dataset from dynamic motions on a cadaver. In contrast with existing references, this enables a true model evaluation that overcomes influences from soft-tissue artifacts, orientation and manual palpation errors. This dataset comprises extensive dynamic movements that are recorded with multimodal measurements including trajectories of optical and virtual (via computed tomography) anatomical markers, reference kinematics, inertial measurements, transformation matrices and visualization tools. The dataset can be used either as a ground-truth reference or to advance research in inertial-sensor-to-bone-alignment.

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We present a novel approach to estimate the relative sensor orientation from inertial sensors placed on connected body segments. Drift in the relative orientation estimates obtained by integrating the gyroscope measurements is corrected solely by incorporating common information in the inertial sensor measurements due to the connection of the body segments. We solve the estimation problem using a complementary filtering implementation to reduce the computational complexity. We study its robustness under common real-world model violations, e.g., soft tissue artifacts and spikes in the acceleration signals due to impacts. The efficacy of the method is illustrated with numerical simulations and is compared to a multiplicative extended Kalman filter implementation, both with and without outlier rejection. In addition, a human experiment strengthened the simulation results under realistic sensor errors.@en