Showing papers on "Kalman filter published in 2009"

Cubature Kalman Filters

Abstract: 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

The probabilistic data association filter

Abstract: The measurement selection for updating the state estimate of a target's track, known as data association, is essential for good performance in the presence of spurious measurements or clutter. A classification of tracking and data association approaches has been presented, as a pure MMSE approach, which amounts to a soft decision, and single best-hypothesis approach, which amounts to a hard decision. It has been shown that the optimal state estimator in the presence of data association uncertainty consists of the computation of the conditional pdf of the state x(k) given all information available at time k, namely, the prior information about the initial state, the intervening known inputs, and the sets of measurements through time k. It has also been pointed out that if the exact conditional pdf, which is a mixture, is available, then its recursion requires only the probabilities of the most recent association events. The conditions under which this result holds, namely whiteness of the noise, detection, and clutter processes, were presented. The PDAF and JPDAF algorithms, which carry out data association and state estimation in clutter, have been described. A simple example was given to illustrate how the clutter and occasional missed detections can lead to track loss for a standard tracking filter, and how PDAF can keep the target in track under such circumstances. By using the Monte Carlo in a simulated based surveillance as an exampled shown. The numerous applications of the PDAF/JPDAF illustrated in "Real-World Applications of PDAF and JPDAF" show the potential pitfalls of using sophisticated algorithms for tracking in difficult environments as well as how to overcome them. The effect of finite sensor resolution can be a more severe problem than the data association and deserves special attention.

Kalman-Consensus Filter : Optimality, stability, and performance

Abstract: One of the fundamental problems in sensor networks is to estimate and track the state of targets (or dynamic processes) of interest that evolve in the sensing field. Kalman filtering has been an effective algorithm for tracking dynamic processes for over four decades. Distributed Kalman Filtering (DKF) involves design of the information processing algorithm of a network of estimator agents with a two-fold objective: 1) estimate the state of the target of interest and 2) reach a consensus with neighboring estimator agents on the state estimate. We refer to this DKF algorithm as Kalman-Consensus Filter (KCF). The main contributions of this paper are as follows: i) finding the optimal decentralized Kalman-Consensus filter and showing that its computational and communication costs are not scalable in n and ii) introducing a scalable suboptimal Kalman-Consensus Filter and providing a formal stability and performance analysis of this distributed and cooperative filtering algorithm. Kalman-Consensus Filtering algorithm is applicable to sensor networks with variable topology including mobile sensor networks and networks with packet-loss.

A comparative survey of WLAN location fingerprinting methods

Abstract: The term “location fingerprinting” covers a wide variety of methods for determining receiver position using databases of radio signal strength measurements from different sources. In this work we present a survey of location fingerprinting methods, including deterministic and probabilistic methods for static estimation, as well as filtering methods based on Bayesian filter and Kalman filter. We present a unified mathematical formulation of radio map database and location estimation, point out the equivalence of some methods from the literature, and present some new variants. A set of tests in an indoor positioning scenario using WLAN signal strengths is performed to determine the influence of different calibration and location method parameters. In the tests, the probabilistic method with the kernel function approximation of signal strength histograms was the best static positioning method. Moreover, all filters improved the results significantly over the static methods.

Recursive Noise Adaptive Kalman Filtering by Variational Bayesian Approximations

Abstract: This article considers the application of variational Bayesian methods to joint recursive estimation of the dynamic state and the time-varying measurement noise parameters in linear state space models. The proposed adaptive Kalman filtering method is based on forming a separable variational approximation to the joint posterior distribution of states and noise parameters on each time step separately. The result is a recursive algorithm, where on each step the state is estimated with Kalman filter and the sufficient statistics of the noise variances are estimated with a fixed-point iteration. The performance of the algorithm is demonstrated with simulated data.

Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models

Abstract: This book was written in response to the growing demand for a text that provides a unified treatment of linear and nonlinear complex valued adaptive filters, and methods for the processing of general complex signals (circular and noncircular). It brings together adaptive filtering algorithms for feedforward (transversal) and feedback architectures and the recent developments in the statistics of complex variable, under the powerful frameworks of CR (Wirtinger) calculus and augmented complex statistics. This offers a number of theoretical performance gains, which is illustrated on both stochastic gradient algorithms, such as the augmented complex least mean square (ACLMS), and those based on Kalman filters. This work is supported by a number of simulations using synthetic and real world data, including the noncircular and intermittent radar and wind signals.

The ensemble Kalman filter for combined state and parameter estimation

Abstract: This article provides a fundamental theoretical basis for understanding EnKF and serves as a useful text for future users. Data assimilation and parameter-estimation problems are explained, and the concept of joint parameter and state estimation, which can be solved using ensemble methods, is presented. KF and EKF are briefly discussed before introducing and deriving EnKF. Similarities and differences between KF and EnKF are pointed out. The benefits of using EnKF with high-dimensional and highly nonlinear dynamical models are illustrated by examples. EnKF and EnKS are also derived from Bayes theorem, using a probabilistic approach. The derivation is based on the assumption that measurement errors are independent in time and the model represents a Markov process, which allows for Bayes theorem to be written in a recursive form, where measurements are processed sequentially in time. The practical implementation of the analysis scheme isdiscussed, and it is shown that it can be computed efficiently in the space spanned by the ensemble realizations. The square root scheme is discussed as an alternative method that avoids the perturbation of measurements. However, the square root scheme has other pitfalls, and it is recommended to use the symmetric square root with or without a random rotation. The random rotation introduces a stochastic component to the update, and the quality of the scheme may then not improve compared to the original stochastic EnKF scheme with perturbed measurements.

Dynamic Linear Models with R

Abstract: Introduction: basic notions about Bayesian inference.- Dynamic linear models.- Model specification.- Models with unknown parameters.- Sequential Monte Carlo methods.

The unscented Kalman filter and particle filter methods for nonlinear structural system identification with non-collocated heterogeneous sensing†

Abstract: The use of heterogeneous, non-collocated measurements for nonlinear structural system identification is explored herein. In particular, this paper considers the example of sensor heterogeneity arising from the fact that both acceleration and displacement are measured at various locations of the structural system. The availability of non-collocated data might often arise in the identification of systems where the displacement data may be provided through global positioning systems (GPS). The well-known extended Kalman filter (EKF) is often used to deal with nonlinear system identification. However, as suggested in (J. Eng. Mech. 1999; 125(2):133–142), the EKF is not effective in the case of highly nonlinear problems. Instead, two techniques are examined herein, the unscented Kalman filter method (UKF), proposed by Julier and Uhlman, and the particle filter method, also known as sequential Monte Carlo method (SMC). The two methods are compared and their efficiency is evaluated through the example of a three degree-of-freedom system, involving a Bouc–Wen hysteretic component, where the availability of displacement and acceleration measurements for different DOFs is assumed. Copyright © 2008 John Wiley & Sons, Ltd.

Vision-Aided Inertial Navigation for Spacecraft Entry, Descent, and Landing

Abstract: In this paper, we present the vision-aided inertial navigation (VISINAV) algorithm that enables precision planetary landing. The vision front-end of the VISINAV system extracts 2-D-to-3-D correspondences between descent images and a surface map (mapped landmarks), as well as 2-D-to-2-D feature tracks through a sequence of descent images (opportunistic features). An extended Kalman filter (EKF) tightly integrates both types of visual feature observations with measurements from an inertial measurement unit. The filter computes accurate estimates of the lander's terrain-relative position, attitude, and velocity, in a resource-adaptive and hence real-time capable fashion. In addition to the technical analysis of the algorithm, the paper presents validation results from a sounding-rocket test flight, showing estimation errors of only 0.16 m/s for velocity and 6.4 m for position at touchdown. These results vastly improve current state of the art for terminal descent navigation without visual updates, and meet the requirements of future planetary exploration missions.

GP-BayesFilters: Bayesian filtering using Gaussian process prediction and observation models

Abstract: Bayesian filtering is a general framework for recursively estimating the state of a dynamical system. Key components of each Bayes filter are probabilistic prediction and observation models. This paper shows how non-parametric Gaussian process (GP) regression can be used for learning such models from training data. We also show how Gaussian process models can be integrated into different versions of Bayes filters, namely particle filters and extended and unscented Kalman filters. The resulting GP-BayesFilters can have several advantages over standard (parametric) filters. Most importantly, GP-BayesFilters do not require an accurate, parametric model of the system. Given enough training data, they enable improved tracking accuracy compared to parametric models, and they degrade gracefully with increased model uncertainty. These advantages stem from the fact that GPs consider both the noise in the system and the uncertainty in the model. If an approximate parametric model is available, it can be incorporated into the GP, resulting in further performance improvements. In experiments, we show different properties of GP-BayesFilters using data collected with an autonomous micro-blimp as well as synthetic data.

Distant Speech Recognition

Abstract: Foreword. Preface. 1 Introduction. 1.1 Research and Applications in Academia and Industry. 1.2 Challenges in Distant Speech Recognition. 1.3 System Evaluation. 1.4 Fields of Speech Recognition. 1.5 Robust Perception. 1.6 Organizations, Conferences and Journals. 1.7 Useful Tools, Data Resources and Evaluation Campaigns. 1.8 Organization of this Book. 1.9 Principal Symbols used Throughout the Book. 1.10 Units used Throughout the Book. 2 Acoustics. 2.1 Physical Aspect of Sound. 2.2 Speech Signals. 2.3 Human Perception of Sound. 2.4 The Acoustic Environment. 2.5 Recording Techniques and Sensor Configuration. 2.6 Summary and Further Reading. 2.7 Principal Symbols. 3 Signal Processing and Filtering Techniques. 3.1 Linear Time-Invariant Systems. 3.2 The Discrete Fourier Transform. 3.3 Short-Time Fourier Transform. 3.4 Summary and Further Reading. 3.5 Principal Symbols. 4 Bayesian Filters. 4.1 Sequential Bayesian Estimation. 4.2 Wiener Filter. 4.3 Kalman Filter and Variations. 4.4 Particle Filters. 4.5 Summary and Further Reading. 4.6 Principal Symbols. 5 Speech Feature Extraction. 5.1 Short-Time Spectral Analysis. 5.2 Perceptually Motivated Representation. 5.3 Spectral Estimation and Analysis. 5.4 Cepstral Processing. 5.5 Comparison between Mel Frequency, Perceptual LP and warped MVDR Cepstral Coefficient Frontends. 5.6 Feature Augmentation. 5.7 Feature Reduction. 5.8 Feature-Space Minimum Phone Error. 5.9 Summary and Further Reading. 5.10 Principal Symbols. 6 Speech Feature Enhancement. 6.1 Noise and Reverberation in Various Domains. 6.2 Two Principal Approaches. 6.3 Direct Speech Feature Enhancement. 6.4 Schematics of Indirect Speech Feature Enhancement. 6.5 Estimating Additive Distortion. 6.6 Estimating Convolutional Distortion. 6.7 Distortion Evolution. 6.8 Distortion Evaluation. 6.9 Distortion Compensation. 6.10 Joint Estimation of Additive and Convolutional Distortions. 6.11 Observation Uncertainty. 6.12 Summary and Further Reading. 6.13 Principal Symbols. 7 Search: Finding the Best Word Hypothesis. 7.1 Fundamentals of Search. 7.2 Weighted Finite-State Transducers. 7.3 Knowledge Sources. 7.4 Fast On-the-Fly Composition. 7.5 Word and Lattice Combination. 7.6 Summary and Further Reading. 7.7 Principal Symbols. 8 Hidden Markov Model Parameter Estimation. 8.1 Maximum Likelihood Parameter Estimation. 8.2 Discriminative Parameter Estimation. 8.3 Summary and Further Reading. 8.4 Principal Symbols. 9 Feature and Model Transformation. 9.1 Feature Transformation Techniques. 9.2 Model Transformation Techniques. 9.3 Acoustic Model Combination. 9.4 Summary and Further Reading. 9.5 Principal Symbols. 10 Speaker Localization and Tracking. 10.1 Conventional Techniques. 10.2 Speaker Tracking with the Kalman Filter. 10.3 Tracking Multiple Simultaneous Speakers. 10.4 Audio-Visual Speaker Tracking. 10.5 Speaker Tracking with the Particle Filter. 10.6 Summary and Further Reading. 10.7 Principal Symbols. 11 Digital Filter Banks. 11.1 Uniform Discrete Fourier Transform Filter Banks. 11.2 Polyphase Implementation. 11.3 Decimation and Expansion. 11.4 Noble Identities. 11.5 Nyquist( M ) Filters. 11.6 Filter Bank Design of De Haan et al . 11.7 Filter Bank Design with the Nyquist( M ) Criterion. 11.8 Quality Assessment of Filter Bank Prototypes. 11.9 Summary and Further Reading. 11.10 Principal Symbols. 12 Blind Source Separation. 12.1 Channel Quality and Selection. 12.2 Independent Component Analysis. 12.3 BSS Algorithms based on Second-Order Statistics. 12.4 Summary and Further Reading. 12.5 Principal Symbols. 13 Beamforming. 13.1 Beamforming Fundamentals. 13.2 Beamforming Performance Measures. 13.3 Conventional Beamforming Algorithms. 13.4 Recursive Algorithms. 13.5 Nonconventional Beamforming Algorithms. 13.6 Array Shape Calibration. 13.7 Summary and Further Reading. 13.8 Principal Symbols. 14 Hands On. 14.1 Example Room Configurations. 14.2 Automatic Speech Recognition Engines. 14.3 Word Error Rate. 14.4 Single-Channel Feature Enhancement Experiments. 14.5 Acoustic Speaker-Tracking Experiments. 14.6 Audio-Video Speaker-Tracking Experiments. 14.7 Speaker-Tracking Performance vs Word Error Rate. 14.8 Single-Speaker Beamforming Experiments. 14.9 Speech Separation Experiments. 14.10 Filter Bank Experiments. 14.11 Summary and Further Reading. Appendices. A List of Abbreviations. B Useful Background. B.1 Discrete Cosine Transform. B.2 Matrix Inversion Lemma. B.3 Cholesky Decomposition. B.4 Distance Measures. B.5 Super-Gaussian Probability Density Functions. B.6 Entropy. B.7 Relative Entropy. B.8 Transformation Law of Probabilities. B.9 Cascade of Warping Stages. B.10 Taylor Series. B.11 Correlation and Covariance. B.12 Bessel Functions. B.13 Proof of the Nyquist-Shannon Sampling Theorem. B.14 Proof of Equations (11.31-11.32). B.15 Givens Rotations. B.16 Derivatives with Respect to Complex Vectors. B.17 Perpendicular Projection Operators. Bibliography. Index.

RSSI-Based Indoor Localization and Tracking Using Sigma-Point Kalman Smoothers

Abstract: Solutions for indoor tracking and localization have become more critical with recent advancement in context and location-aware technologies. The accuracy of explicit positioning sensors such as global positioning system (GPS) is often limited for indoor environments. In this paper, we evaluate the feasibility of building an indoor location tracking system that is cost effective for large scale deployments, can operate over existing Wi-Fi networks, and can provide flexibility to accommodate new sensor observations as they become available. This paper proposes a sigma-point Kalman smoother (SPKS)-based location and tracking algorithm as a superior alternative for indoor positioning. The proposed SPKS fuses a dynamic model of human walking with a number of low-cost sensor observations to track 2-D position and velocity. Available sensors include Wi-Fi received signal strength indication (RSSI), binary infra-red (IR) motion sensors, and binary foot-switches. Wi-Fi signal strength is measured using a receiver tag developed by Ekahau, Inc. The performance of the proposed algorithm is compared with a commercially available positioning engine, also developed by Ekahau, Inc. The superior accuracy of our approach over a number of trials is demonstrated.

Dynamic linear models

Abstract: In this chapter we discuss the basic notions about state space models and their use in time series analysis. The dynamic linear model is presented as a special case of a general state space model, being linear and Gaussian. For dynamic linear models, estimation and forecasting can be obtained recursively by the well-known Kalman filter.

Distributed Kriged Kalman Filter for Spatial Estimation

Abstract: This paper considers robotic sensor networks performing spatially-distributed estimation tasks. A robotic sensor network is deployed in an environment of interest, and takes successive point measurements of a dynamic physical process modeled as a spatio-temporal random field. Taking a Bayesian perspective on the Kriging interpolation technique from geostatistics, we design the distributed Kriged Kalman filter for predictive inference of the random field and of its gradient. The proposed algorithm makes use of a novel distributed strategy to compute weighted least squares estimates when measurements are spatially correlated. This strategy results from the combination of the Jacobi overrelaxation method with dynamic average consensus algorithms. As an application of the proposed algorithm, we design a gradient ascent cooperative strategy and analyze its convergence properties in the absence of measurement errors via stochastic Lyapunov functions. We illustrate our results in simulation.

Structural Time Series Analyser, Modeller and Predictor: STAMP 8.2.

Abstract: STAMP™ stands for Structural Time series Analyser, Modeller and Predictor. It is a menu-driven system designed to model, describe and predict time series. It is based on structural time series models. These models are set up in terms of components such as trends, seasonals and cycles, which have a direct interpretation. Estimation is carried out using state space methods and Kalman filtering. STAMP 8.2 for OxMetrics 6 handles time series with missing values. Explanatory variables with time varying coefficients and interventions can be included. Version 8 includes extensions and improvements for Multivariate Models: select components by equation, select regressors and interventions by equation, separate dependence structures for each component, wide choice of variance matrices, higher order multivariate components, missing observations allowed, forecasting, exact likelihood computation, automatic outlier and break detection, fixing parameters is made easy. Among the special features of STAMP are interactive model selection, a wide range of diagnostics, easy creation of model based forecasts, spectral filters, observation weight functions, and batch facilities. The STAMP book introduces structural time series models and the way in which they can be used to model a wide range of series.

Explicit Model Predictive Control of DC–DC Switched-Mode Power Supplies With Extended Kalman Filtering

Abstract: This paper presents a sensorless explicit model predictive control scheme for the dc-dc boost converter. No direct inductor current measurement is needed as the coil current is derived either via a static approximation or, for improved accuracy, through an extended Kalman filter. The estimate is used in the chosen optimal control problem formulation which yields the optimal input by intrinsically accounting for duty cycle and current constraints. The optimization problem is explicitly presolved offline so that the online effort is reduced to a simple search in the resulting lookup table. No online optimization is required, greatly facilitating physical implementation and allowing for experimental validation on an integrated dc-dc converter through a fixed-point DSP.

Efficient simulation and integrated likelihood estimation in state space models

Abstract: We consider the problem of implementing simple and efficient Markov chain Monte Carlo (MCMC) estimation algorithms for state space models. A conceptually transparent derivation of the posterior distribution of the states is discussed, which also leads to an efficient simulation algorithm that is modular, scalable and widely applicable. We also discuss a simple approach for evaluating the integrated likelihood, defined as the density of the data given the parameters but marginal of the state vector. We show that this high-dimensional integral can be easily evaluated with minimal computational and conceptual difficulty. Two empirical applications in macroeconomics demonstrate that the methods are versatile and computationally undemanding. In one application, involving a time-varying parameter model, we show that the methods allow for efficient handling of large state vectors. In our second application, involving a dynamic factor model, we introduce a new blocking strategy which results in improved MCMC mixing at little cost. The results demonstrate that the framework is simple, flexible and efficient.

New Battery Model and State-of-Health Determination Through Subspace Parameter Estimation and State-Observer Techniques

Abstract: This paper describes a novel adaptive battery model based on a remapped variant of the well-known Randles' lead-acid model. Remapping of the model is shown to allow improved modeling capabilities and accurate estimates of dynamic circuit parameters when used with subspace parameter-estimation techniques. The performance of the proposed methodology is demonstrated by application to batteries for an all-electric personal rapid transit vehicle from the urban light transport (ULTRA) program, which is designated for use at Heathrow Airport, U.K. The advantages of the proposed model over the Randles' circuit are demonstrated by comparisons with alternative observer/estimator techniques, such as the basic Utkin observer and the Kalman estimator. These techniques correctly identify and converge on voltages associated with the battery state-of-charge (SoC), despite erroneous initial conditions, thereby overcoming problems attributed to SoC drift (incurred by Coulomb-counting methods due to overcharging or ambient temperature fluctuations). Observation of these voltages, as well as online monitoring of the degradation of the estimated dynamic model parameters, allows battery aging (state-of-health) to also be assessed and, thereby, cell failure to be predicted. Due to the adaptive nature of the proposed algorithms, the techniques are suitable for applications over a wide range of operating environments, including large ambient temperature variations. Moreover, alternative battery topologies may also be accommodated by the automatic adjustment of the underlying state-space models used in both the parameter-estimation and observer/estimator stages.

Kalman filtering for positioning and heading control of ships and offshore rigs

Abstract: In this article, we have described the main components of a ship motion-control system and two particular motion-control problems that require wave filtering, namely, dynamic positioning and heading autopilot. Then, we discussed the models commonly used for vessel response and showed how these models are used for Kalman filter design. We also briefly discussed parameter and noise covariance estimation, which are used for filter tuning. To illustrate the performance, a case study based on numerical simulations for a ship autopilot was considered. The material discussed in this article conforms to modern commercially available ship motion-control systems. Most of the vessels operating in the offshore industry worldwide use Kalman filters for velocity estimation and wave filtering. Thus, the article provides an up-to-date tutorial and overview of Kalman-filter-based wave filtering.

Unscented Kalman Filter for Brain-Machine Interfaces

Abstract: Brain machine interfaces (BMIs) are devices that convert neural signals into commands to directly control artificial actuators, such as limb prostheses. Previous real-time methods applied to decoding behavioral commands from the activity of populations of neurons have generally relied upon linear models of neural tuning and were limited in the way they used the abundant statistical information contained in the movement profiles of motor tasks. Here, we propose an n-th order unscented Kalman filter which implements two key features: (1) use of a non-linear (quadratic) model of neural tuning which describes neural activity significantly better than commonly-used linear tuning models, and (2) augmentation of the movement state variables with a history of n-1 recent states, which improves prediction of the desired command even before incorporating neural activity information and allows the tuning model to capture relationships between neural activity and movement at multiple time offsets simultaneously. This new filter was tested in BMI experiments in which rhesus monkeys used their cortical activity, recorded through chronically implanted multielectrode arrays, to directly control computer cursors. The 10th order unscented Kalman filter outperformed the standard Kalman filter and the Wiener filter in both off-line reconstruction of movement trajectories and real-time, closed-loop BMI operation.

Structural time series models and the Kalman filter: A concise review

Abstract: The continued increase in availability of economic data in recent years and, more importantly, the possibility to construct larger frequency time series, have fostered the use (and development) of statistical and econometric techniques to treat them more accurately. This paper presents an exposition of structural time series models by which a time series can be decomposed as the sum of a trend, seasonal and irregular components. In addition to a detailled analysis of univariate speci cations we also address the SUTSE multivariate case and the issue of cointegration. Finally, the recursive estimation and smoothing by means of the Kalman lter algorithm is described taking into account its different stages, from initialisation to parameter's estimation.

Invariant Extended Kalman Filter: theory and application to a velocity-aided attitude estimation problem

Abstract: A new version of the Extended Kalman Filter (EKF) is proposed for nonlinear systems possessing symmetries. Instead of using a linear correction term based on a linear output error, it uses a geometrically adapted correction term based on an invariant output error; in the same way the gain matrix is not updated from of a linear state error, but from an invariant state error. The benefit is that the gain and covariance equations converge to constant values on a much bigger set of trajectories than equilibrium points as is the case for the EKF, which should result in a better convergence of the estimation. This filter is applied to the practically relevant problem of estimating the velocity and attitude of a moving rigid body, e.g. an aircraft, from GPS velocity, inertial and magnetic measurements. In this context it can be seen as an extension of the “Multiplicative EKF” often used for quaternion estimation.

Robust lane detection and tracking with ransac and Kalman filter

Abstract: In a previous paper, a simple approach to lane detection using the Hough transform and iterated matched filters was described [1]. This paper extends this work by incorporating an inverse perspective mapping to create a bird's-eye view of the road, applying random sample consensus to help eliminate outliers due to noise and artifacts in the road, and a Kalman filter to help smooth the output of the lane tracker.

Kalman Filtering with Nonlinear State Constraints

Abstract: An analytic method was developed by D. Simon and T. L. Chia to incorporate linear state equality constraints into the Kalman filter. When the state constraint was nonlinear, linearization was employed to obtain an approximately linear constraint around the current state estimate. This linearized constrained Kalman filter is subject to approximation errors and may suffer from a lack of convergence. We present a method that allows exact use of second-order nonlinear state constraints. It is based on a computational algorithm that iteratively finds the Lagrangian multiplier for the nonlinear constraints. Computer simulation results are presented to illustrate the algorithm.

Performance Analysis of Vector Tracking Algorithms for Weak GPS Signals in High Dynamics

Abstract: This paper explores the ability of vector tracking algorithms to track weak Global Positioning System (GPS) signals in high dynamic environments. Traditional GPS receivers use tracking loops to track the GPS signals. The signals from each satellite are processed independently. In contrast, vector-based methods do not use tracking loops. Instead, all the satellite signals are tracked by a lone Kalman filter. The Kalman filter combines the tasks of signal tracking and navigation into a single algorithm. Vector-based methods can perform better than traditional methods in environments with high dynamics and low signal power. A performance analysis of the vector tracking algorithms is included. The ability of the algorithms to operate as a function of carrier to noise power density ratio, user dynamics, and number of satellites being used is explored. The vector tracking methods are demonstrated using data from a high fidelity GPS simulator. The simulation results show the vector tracking algorithms operating at a carrier to noise power density ratio of 19 dB-Hz through 2 G, 4 G, and 8 G coordinated turns. The vector tracking algorithms are also shown operating through 2 G and 4 G turns at a carrier to noise power density ratio of 16 dB-Hz.