An introduction to the Kalman filter

In 1960, R.E. Kalman published his famous paper describing a recursive solution to the discrete-data linear filtering problem. Since that time, due in large part to advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. The Kalman filter is a set of mathematical equations that provides an efficient computational (recursive) means to estimate the state of a process, in a way that minimizes the mean of the squared error. The filter is very powerful in several aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown. The purpose of this paper is to provide a practical introduction to the discrete Kalman filter. This introduction includes a description and some discussion of the basic discrete Kalman filter, a derivation, description and some discussion of the extended Kalman filter, and a relatively simple (tangible) example with real numbers & results.

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An Introduction to the Kalman Filter

In this paper, we present a new nonlinear filter for high-dimensional state estimation, which we have named the cubature Kalman filter (CKF) The heart of the CKF is a spherical-radial cubature rule, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear Bayesian filter Specifically, we derive a third-degree spherical-radial cubature rule that provides a set of cubature points scaling linearly with the state-vector dimension The CKF may therefore provide a systematic solution for high-dimensional nonlinear filtering problems The paper also includes the derivation of a square-root version of the CKF for improved numerical stability The CKF is tested experimentally in two nonlinear state estimation problems In the first problem, the proposed cubature rule is used to compute the second-order statistics of a nonlinearly transformed Gaussian random variable The second problem addresses the use of the CKF for tracking a maneuvering aircraft The results of both experiments demonstrate the improved performance of the CKF over conventional nonlinear filters

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Cubature Kalman Filters

The technology of hydrodynamic modeling and marine craft motion control systems has progressed greatly in recent years. This timely survey includes the latest tools for analysis and design of advanced guidance, navigation and control systems and presents new material on underwater vehicles and surface vessels. Each section presents numerous case studies and applications, providing a practical understanding of how model-based motion control systems are designed.

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Handbook of Marine Craft Hydrodynamics and Motion Control: Fossen/Handbook of Marine Craft Hydrodynamics and Motion Control

Now in its second edition, this accessible text presents a unified Bayesian treatment of state-of-the-art filtering, smoothing, and parameter estimation algorithms for non-linear state space models. The book focuses on discrete-time state space models and carefully introduces fundamental aspects related to optimal filtering and smoothing. In particular, it covers a range of efficient non-linear Gaussian filtering and smoothing algorithms, as well as Monte Carlo-based algorithms. This updated edition features new chapters on constructing state space models of practical systems, the discretization of continuous-time state space models, Gaussian filtering by enabling approximations, posterior linearization filtering, and the corresponding smoothers. Coverage of key topics is expanded, including extended Kalman filtering and smoothing, and parameter estimation. The book's practical, algorithmic approach assumes only modest mathematical prerequisites, suitable for graduate and advanced undergraduate students. Many examples are included, with Matlab and Python code available online, enabling readers to implement algorithms in their own projects.

Bayesian Filtering and Smoothing

The definitive textbook and professional reference on Kalman Filtering fully updated, revised, and expanded This book contains the latest developments in the implementation and application of Kalman filtering. Authors Grewal and Andrews draw upon their decades of experience to offer an in-depth examination of the subtleties, common pitfalls, and limitations of estimation theory as it applies to real-world situations. They present many illustrative examples including adaptations for nonlinear filtering, global navigation satellite systems, the error modeling of gyros and accelerometers, inertial navigation systems, and freeway traffic control. Kalman Filtering: Theory and Practice Using MATLAB, Fourth Edition is an ideal textbook in advanced undergraduate and beginning graduate courses in stochastic processes and Kalman filtering. It is also appropriate for self-instruction or review by practicing engineers and scientists who want to learn more about this important topic.

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Kalman Filtering: Theory and Practice Using MATLAB

An updated guide to GNSS and INS, and solutions to real-world GPS/INS problems with Kalman filtering Written by recognized authorities in the field, this second edition of a landmark work provides engineers, computer scientists, and others with a working familiarity with the theory and contemporary applications of Global Navigation Satellite Systems (GNSS), Inertial Navigational Systems (INS), and Kalman filters. Throughout, the focus is on solving real-world problems, with an emphasis on the effective use of state-of-the-art integration techniques for those systems, especially the application of Kalman filtering. To that end, the authors explore the various subtleties, common failures, and inherent limitations of the theory as it applies to real-world situations, and provide numerous detailed application examples and practice problems, including GNSS-aided INS, modeling of gyros and accelerometers, and SBAS and GBAS. Drawing upon their many years of experience with GNSS, INS, and the Kalman filter, the authors present numerous design and implementation techniques not found in other professional references. This Second Edition has been updated to include: GNSS signal integrity with SBAS Mitigation of multipath, including results Ionospheric delay estimation with Kalman filters New MATLAB programs for satellite position determination using almanac and ephemeris data and ionospheric delay calculations from single and dual frequency data New algorithms for GEO with L1 /L5 frequencies and clock steering Implementation of mechanization equations in numerically stable algorithms To enhance comprehension of the subjects covered, the authors have included software in MATLAB, demonstrating the working of the GNSS, INS, and filter algorithms. In addition to showing the Kalman filter in action, the software also demonstrates various practical aspects of finite word length arithmetic and the need for alternative algorithms to preserve result accuracy.

Global Positioning Systems, Inertial Navigation, and Integration

1. General Information. 2. Linear Dynamic Systems. 3. Random Processes and Stochastic Systems. 4. Linear Optimal Filters and Predictors. 5. Nonlinear Applications. 6. Implementation Methods. 7. Practical Considerations. Appendix A: Software. Appendix B: A Matrix Refresher.

Kalman Filtering: Theory and Practice

An updated guide to GNSS, and INS, and solutions to real-world GNSS/INS problems with Kalman filteringWritten by recognized authorities in the field, this third edition of a landmark work provides engineers, computer scientists, and others with a working familiarity of the theory and contemporary applications of Global Navigation Satellite Systems (GNSS), Inertial Navigational Systems, and Kalman filters. Throughout, the focus is on solving real-world problems, with an emphasis on the effective use of state-of-the-art integration techniques for those systems, especially the application of Kalman filtering. To that end, the authors explore the various subtleties, common failures, and inherent limitations of the theory as it applies to real-world situations, and provide numerous detailed application examples and practice problems, including GNSS-aided INS (tightly and loosely coupled), modeling of gyros and accelerometers, and SBAS and GBAS.Drawing upon their many years of experience with GNSS, INS, and the Kalman filter, the authors present numerous design and implementation techniques not found in other professional references. The Third Edition includes:Updates on the upgrades in existing GNSS and other systems currently under developmentExpanded coverage of basic principles of antenna design and practical antenna design solutionsExpanded coverage of basic principles of receiver design and an update of the foundations for code and carrier acquisition and tracking within a GNSS receiverExpanded coverage of inertial navigation, its history, its technology, and the mathematical models and methods used in its implementationDerivations of dynamic models for the propagation of inertial navigation errors, including the effects of drifting sensor compensation parametersGreatly expanded coverage of GNSS/INS integration, including derivation of a unified GNSS/INS integration model, its MATLAB implementations, and performance evaluation under simulated dynamic conditionsThe companion website includes updated background material; additional MATLAB scripts for simulating GNSS-only and integrated GNSS/INS navigation; satellite position determination; calculation of ionosphere delays; and dilution of precision.

Global Navigation Satellite Systems, Inertial Navigation, and Integration

The methodology of discrete time, extended Kalman filtering is applied to the problem of identifying parameters of a macroscopic freeway traffic model. Macroscopic models provide a representation of traffic flow in terms of its gross properties, i.e., volume, density, and speed. The local identifiability of a parameterization of macroscopic model at nominal values of the unknown parameters is checked before any identification is attempted. It is shown that the parameterization is locally identifiable. Two parameters of the model (reaction time and sensitivity to changing density) were identified through the use of this methodology. The data base for studies to date was generated from a microscopic simulation of freeway traffic, which involves following all individual vehicle movements. Techniques for extending the methodology to employ real freeway traffic data, especially as can be obtained from automated surveillance systems, are discussed.

Mohinder S. Grewal

Papers

Kalman Filtering: Theory and Practice Using MATLAB

Global Positioning Systems, Inertial Navigation, and Integration

Kalman Filtering: Theory and Practice

Global Navigation Satellite Systems, Inertial Navigation, and Integration

Identification of Parameters in a Freeway Traffic Model