Gustaf Hendeby

Michael Roth, Gustaf Hendeby, Carsten Fritsche et al.

The ensemble Kalman filter (EnKF) is a Monte Carlo based implementation of the Kalman filter (KF) for extremely high-dimensional, possibly nonlinear and non-Gaussian state estimation problems. Its ability to handle state dimensions in the order of millions has made the EnKF a popular algorithm in different geoscientific disciplines. Despite a similarly vital need for scalable algorithms in signal processing, e.g., to make sense of the ever increasing amount of sensor data, the EnKF is hardly discussed in our field. This self-contained review paper is aimed at signal processing researchers and provides all the knowledge to get started with the EnKF. The algorithm is derived in a KF framework, without the often encountered geoscientific terminology. Algorithmic challenges and required extensions of the EnKF are provided, as well as relations to sigma-point KF and particle filters. The relevant EnKF literature is summarized in an extensive survey and unique simulation examples, including popular benchmark problems, complement the theory with practical insights. The signal processing perspective highlights new directions of research and facilitates the exchange of potentially beneficial ideas, both for the EnKF and high-dimensional nonlinear and non-Gaussian filtering in general.

Anton Kullberg, Frida Viset, Isaac Skog et al.

Basis Function (BF) expansions are a cornerstone of any engineer's toolbox for computational function approximation which shares connections with both neural networks and Gaussian processes. Even though BF expansions are an intuitive and straightforward model to use, they suffer from quadratic computational complexity in the number of BFs if the predictive variance is to be computed. We develop a method to automatically select the most important BFs for prediction in a sub-domain of the model domain. This significantly reduces the computational complexity of computing predictions while maintaining predictive accuracy. The proposed method is demonstrated using two numerical examples, where reductions up to 50-75% are possible without significantly reducing the predictive accuracy.

Chuan Huang, Gustaf Hendeby, Isaac Skog

Spatially inhomogeneous magnetic fields offer a valuable, non-visual information source for positioning. Among systems leveraging this, magnetic field-based simultaneous localization and mapping (SLAM) systems are particularly attractive. These systems execute positioning and magnetic field mapping tasks simultaneously, and they have bounded positioning error within previously visited regions. However, state-of-the-art magnetic-field SLAM methods typically require low-drift odometry data provided by visual odometry, a wheel encoder, or pedestrian dead-reckoning technology. To address this limitation, this work proposes loosely coupled and tightly coupled inertial magnetic SLAM (IM-SLAM) systems, which use only low-cost sensors: an inertial measurement unit (IMU), 30 magnetometers, and a barometer. Both systems are based on a magnetic-field-aided inertial navigation system (INS) and use error-state Kalman filters for state estimation. The key difference between the two systems is whether the navigation state estimation is done in one or two steps. These systems are evaluated in real-world indoor environments with multi-floor structures. The results of the experiment show that the tightly coupled IM-SLAM system achieves lower positioning errors than the loosely coupled system in most scenarios, with typical errors on the order of meters per 100 meters traveled. These results demonstrate the feasibility of developing a full 3D IM-SLAM system using low-cost sensors. A potential application of the proposed systems is for the positioning of emergency response officers.

Chuan Huang, Gustaf Hendeby, Isaac Skog

This paper presents a new approach for jointly calibrating magnetometers and inertial measurement units, focusing on improving calibration accuracy and computational efficiency. The proposed method formulates the calibration problem as a maximum a posteriori estimation problem, treating both the calibration parameters and orientation trajectory of the sensors as unknowns. This formulation enables efficient optimization with closed-form derivatives. The method is compared against two state-of-the-art approaches in terms of computational complexity and estimation accuracy. Simulation results demonstrate that the proposed method achieves lower root mean square error in calibration parameters while maintaining competitive computational efficiency. Further validation through real-world experiments confirms the practical benefits of our approach: it effectively reduces position drift in a magnetic field-aided inertial navigation system by more than a factor of two on most datasets. Moreover, the proposed method calibrated 30 magnetometers in less than 2 minutes. The contributions include a new calibration method, an analysis of existing methods, and a comprehensive empirical evaluation. Datasets and algorithms are made publicly available to promote reproducible research.

Håkan Carlsson, Isaac Skog, Gustaf Hendeby et al.

We present a comprehensive framework for fusing measurements from multiple and generally placed accelerometers and gyroscopes to perform inertial navigation. Using the angular acceleration provided by the accelerometer array, we show that the numerical integration of the orientation can be done with second-order accuracy, which is more accurate compared to the traditional first-order accuracy that can be achieved when only using the gyroscopes. Since orientation errors are the most significant error source in inertial navigation, improving the orientation estimation reduces the overall navigation error. The practical performance benefit depends on prior knowledge of the inertial sensor array, and therefore we present four different state-space models using different underlying assumptions regarding the orientation modeling. The models are evaluated using a Lie Group Extended Kalman filter through simulations and real-world experiments. We also show how individual accelerometer biases are unobservable and can be replaced by a six-dimensional bias term whose dimension is fixed and independent of the number of accelerometers.

Parinaz Kasebzadeh, Gustaf Hendeby, Fredrik Gustafsson

An approach for computing unique gait signature using measurements collected from body-worn inertial measurement units (IMUs) is proposed. The gait signature represents one full cycle of the human gait, and is suitable for off-line or on-line classification of the gait mode. The signature can also be used to jointly classify the gait mode and the device mode. The device mode identifies how the IMU-equipped device is being carried by the user. The method is based on precise segmentation and resampling of the measured IMU signal, as an initial step, further tuned by minimizing the variability of the obtained signature within each gait cycle. Finally, a Fourier series expansion of the gait signature is introduced which provides a low-dimensional feature vector well suited for classification purposes. The proposed method is evaluated on a large dataset involving several subjects, each one containing two different gait modes and four different device modes. The gait signatures enable a high classification rate for each step cycle.