Showing papers on "Kalman filter published in 1997"

Time series analysis

Abstract: Preface1Difference Equations12Lag Operators253Stationary ARMA Processes434Forecasting725Maximum Likelihood Estimation1176Spectral Analysis1527Asymptotic Distribution Theory1808Linear Regression Models2009Linear Systems of Simultaneous Equations23310Covariance-Stationary Vector Processes25711Vector Autoregressions29112Bayesian Analysis35113The Kalman Filter37214Generalized Method of Moments40915Models of Nonstationary Time Series43516Processes with Deterministic Time Trends45417Univariate Processes with Unit Roots47518Unit Roots in Multivariate Time Series54419Cointegration57120Full-Information Maximum Likelihood Analysis of Cointegrated Systems63021Time Series Models of Heteroskedasticity65722Modeling Time Series with Changes in Regime677A Mathematical Review704B Statistical Tables751C Answers to Selected Exercises769D Greek Letters and Mathematical Symbols Used in the Text786Author Index789Subject Index792

New extension of the Kalman filter to nonlinear systems

Abstract: The Kalman Filter (KF) is one of the most widely used methods for tracking and estimation due to its simplicity, optimality, tractability and robustness. However, the application of the KF to nonlinear systems can be difficult. The most common approach is to use the Extended Kalman Filter (EKF) which simply linearizes all nonlinear models so that the traditional linear Kalman filter can be applied. Although the EKF (in its many forms) is a widely used filtering strategy, over thirty years of experience with it has led to a general consensus within the tracking and control community that it is difficult to implement, difficult to tune, and only reliable for systems which are almost linear on the time scale of the update intervals. In this paper a new linear estimator is developed and demonstrated. Using the principle that a set of discretely sampled points can be used to parameterize mean and covariance, the estimator yields performance equivalent to the KF for linear systems yet generalizes elegantly to nonlinear systems without the linearization steps required by the EKF. We show analytically that the expected performance of the new approach is superior to that of the EKF and, in fact, is directly comparable to that of the second order Gauss filter. The method is not restricted to assuming that the distributions of noise sources are Gaussian. We argue that the ease of implementation and more accurate estimation features of the new filter recommend its use over the EKF in virtually all applications.

A non-divergent estimation algorithm in the presence of unknown correlations

Abstract: This paper addresses the problem of estimation when the cross-correlation in the errors between different random variables are unknown. A new data fusion algorithm, the covariance intersection algorithm (CI), is presented. It is proved that this algorithm yields consistent estimates irrespective of the actual correlations. This property is illustrated in an application of decentralised estimation where it is impossible to consistently use a Kalman filter.

Fundamentals of high accuracy inertial navigation

Abstract: Part 1 Inertial Navigation: Notation, Coordinate Systems and Units Equations of Motion in a Central Force Gravity Field Inertial Instrumentation Calibration Initial Alignment and Attitude Computation Geodetic Variables and Constants Equations of Motion with General Gravity Model. Part 2 Inertial Navigation with Aids: Inertial Navigation with External Measurements Error Equations for the Kalman Filter State Variable Error Models. Part 3 Accuracy Analysis: Accuracy Criteria and Analysis Techniques Error Equations for Calibration, Alignment and Initialization Evaluation of Gravity Model Error Effects. Appendices: Matrix Inverse Formulas LaPlace Transforms Quaternions Associated Legendre Functions Associated Legendre Function Derivatives Procedure for Generating Gravity Disturbance Realizations Procedure for Generating Specific Force Profile.

SCAAT: incremental tracking with incomplete information

Abstract: The Kalman filter provides a powerful mathematical framework within which a minimum mean-square-error estimate of a user's position and orientation can be tracked using a sequence of single sensor observations, as opposed to groups of observations. We refer to this new approach as single-constraint-at-a-time or SCAAT tracking. The method improves accuracy by properly assimilating sequential observations, filtering sensor measurements, and by concurrently autocalibrating mechanical or electrical devices. The method facilitates user motion prediction, multisensor data fusion, and in systems where the observations are only available sequentially it provides estimates at a higher rate and with lower latency than a multiple-constraint approach. Improved accuracy is realized primarily for three reasons. First, the method avoids mathematically treating truly sequential observations as if they were simultaneous. Second, because each estimate is based on the observation of an individual device, perceived error (statistically unusual estimates) can be more directly attributed to the corresponding device. This can be used for concurrent autocalibration which can be elegantly incorporated into the existing Kalman filter. Third, the Kalman filter inherently addresses the effects of noisy device measurements. Beyond accuracy, the method nicely facilitates motion prediction because the Kalman filter already incorporates a model of the user's dynamics, and because it provides smoothed estimates of the user state, including potentially unmeasured elements. Finally, in systems where the observations are only available sequentially, the method can be used to weave together information from individual devices in a very flexible manner, producing a new estimate as soon as each individual observation becomes available, thus facilitating multisensor data fusion and improving the estimate rates and latencies. The most significant aspect of this work is the introduction and exploration of the SCAAT approach to 3D tracking for virtual environments. However I also believe that this work may prove to be of interest to the larger scientific and engineering community in addressing a more general class of tracking and estimation problems.

Multi-Sensor Fusion: Fundamentals and Applications With Software

Abstract: I. INTRODUCTION TO SENSOR FUSION. 1. Introduction. Importance. Sensor Processes. Applications. Summary. Problem Set 1. II. FOUNDATIONS OF SENSOR FUSION. 2. Sensors. Mathematical Description. Use of Multiple Sensors. Construction of Reliable Abstract Sensors From Simple Abstract Sensors. Static and Dynamic Networks. Conclusion. Problem Set 2. 3. Mathematical Tools Used. Algorithms. Linear Algebra. Coordinate Transformations. Rigid Body Motion. Probability. Dependability and Markov Chains. Gaussian Noise. Meta-Heuristics. Summary. Problem Set 3. 4. High-Performance Data Structures: CAD Based. Boundary Representations. Sweep Presentation. CSG - Constructive Solid Geometry. Wire-Frame Models and the Wing-Edge Data Structure. Surface Patches and Contours. Generalized Cylinders. Summary. Problem Set 4. 5. High-Performance Data Structures: Tessellated. Sparse Arrays. Simplex Grids of Non-Uniform Sizes. Grayscale and Color Arrays. Occupancy Grids and HIMM Histogram Maps. Summary. Problem Set 5. 6. High-Performance Data Structures: Trees, and Graphs. 2n Trees. Forest of Quadtrees. Translation Invariant Data Structure. Multi-Dimensional Trees. Graphs of Free Space. Description Trees of Polygons. Range and Interval Trees. Summary. Problem Set 6. 7. High-Performance Data Structures: Functions. Interpolation. Least Squares Estimation. Splines. Bezier Curves and Bi-Cubic Patches. Fourier Transform, Cepstrum and Wavelets. Modal Representation. Summary. Problem Set 7. 8. Representing Ranges and Uncertainty in Data Structures. Explicit Accuracy Bounds. Probability and Dempster-Shafer Methods. Statistics. Fuzzy Sets. Summary. Problem Set 8. III. APPLICATIONS TO SENSOR FUSION. 9. Image Registration for Sensor Fusion. Image Registration Techniques. Problem Statement. Fitness Function. Tabu Search. Genetic Algorithms. Simulated Annealing. Results. Summary. 10. Designing Optimal Sensor Systems within Dependability Bounds. Applications. Dependability Measures. Optimization Model. Exhaustive Search on the Multidimensional Surface. Experimental Results of the Exhaustive Search Algorithm. Heuristic Methods. Summary. 11. Sensor Fusion and Approximate Agreement. Byzantine Generals Problem. Approximate Byzantine Matching. Fusion of Contradictory Sensor Information. Performance Comparison. Hybrid Algorithm. Example 1. Example 2. Summary. 12. Kalman Filtering Applied to a Sensor Fusion Problem. Background. A New Method. A New Technique for Cloud Removal. A Prototype System. Kalman Filter for Scenario 1. Discussion of Results. Summary. 13. Optimal Sensor Fusion Using Range Trees Recursively. Sensors. Redundancy and Associated Errors. Faulty Sensor Averaging Problem. Interval Trees. Algorithm to Find the Optimal Region. Algorithm Complexity. Comparison with Known Methods. Summary. 14. Distributed Dynamic Sensor Fusion. Problem Description. New Paradigm for Distributed Dynamic Sensor Fusion. Robust Agreement Using the Optimal Region. A Comparison with Existing Approaches. Experimental Results. Summary. IV. CASE STUDIES AND CONCLUSION. 15. Sensor Fusion Case Studies. Levels of Sensor Fusion. Types of Sensors Available. Research Trends. Case Studies. Summary. 16. Conclusion. Review. Conclusion. Appendix A. Program Source Code. References. Index483.

Parametrized structure from motion for 3D adaptive feedback tracking of faces

Abstract: A real-time system is described for automatically detecting, modeling and tracking faces in 3D. A closed loop approach is proposed which utilizes structure from motion to generate a 3D model of a face and then feed back the estimated structure to constrain feature tracking in the next frame. The system initializes by using skin classification, symmetry operations, 3D warping and eigenfaces to find a face. Feature trajectories are then computed by SSD or correlation-based tracking. The trajectories are simultaneously processed by an extended Kalman filter to stably recover 3D structure, camera geometry and facial pose. Adaptively weighted estimation is used in this filter by modeling the noise characteristics of the 2D image patch tracking technique. In addition, the structural estimate is constrained by using parametrized models of facial structure (eigen-heads). The Kalman filter's estimate of the 3D state and motion of the face predicts the trajectory of the features which constrains the search space for the next frame in the video sequence. The feature tracking and Kalman filtering closed loop system operates at 25 Hz.

Convergence analysis of the extended Kalman filter used as an observer for nonlinear deterministic discrete-time systems

Abstract: In this paper, convergence analysis of the extended Kalman filter (EKF), when used as an observer for nonlinear deterministic discrete-time systems, is presented. Based on a new formulation of the first-order linearization technique, sufficient conditions to ensure local asymptotic convergence are established. Furthermore, it is shown that the design of the arbitrary matrix plays an important role in enlarging the domain of attraction and then improving the convergence of the modified EKF significantly. The efficiency of this approach, compared to the classical version of the EKF, is shown through a nonlinear identification problem as well as a state and parameter estimation of nonlinear discrete-time systems.

Adaptive tracking of linear time-variant systems by extended RLS algorithms

Abstract: We exploit the one-to-one correspondences between the recursive least-squares (RLS) and Kalman variables to formulate extended forms of the RLS algorithm. Two particular forms of the extended RLS algorithm are considered: one pertaining to a system identification problem and the other pertaining to the tracking of a chirped sinusoid in additive noise. For both of these applications, experiments are presented that demonstrate the tracking superiority of the extended RLS algorithms compared with the standard RLS and least-mean-squares (LMS) algorithms.

Multi-modal tracking of faces for video communications

Abstract: Visual processes to detect and track faces for video compression and transmission. The system is based on an architecture in which a supervisor selects and activates visual processes in cyclic manner. Control of visual processes is made possible by a confidence factor which accompanies each observation. Fusion of results into a unified estimation for tracking is made possible by estimating a covariance matrix with each observation. Visual processes for face tracking are described using blink detection, normalised color histogram matching, and cross correlation (SSD and NCC). Ensembles of visual processes are organised into processing states so as to provide robust tracking. Transition between states is determined by events detected by processes. The result of face detection is fed into recursive estimator (Kalman filter). The output from the estimator drives a PD controller for a pan/tilt/zoom camera. The resulting system provides robust and precise tracking which operates continuously at approximately 20 images per second on a 150 megahertz computer workstation.

Reduced space optimal analysis for historical data sets: 136 years of Atlantic sea surface temperatures

Abstract: A computationally efficient method for analyzing meteorological and oceanographic historical data sets has been developed. The method combines data reduction and least squares optimal estimation. The data reduction involves computing empirical orthogonal functions (EOFs) of the data based on their recent, high-quality portion and using a leading EOF subset as a basis for the analyzed solution and for fitting a first-order linear model of time transitions. We then formulate optimal estimation problems in terms of the EOF projection of the analyzed field to obtain reduced space analogues of the optimal smoother, the Kalman filter, and optimal interpolation techniques. All reduced space algorithms are far cheaper computationally than their full grid prototypes, while their solutions are not necessarily inferior since the sparsity and error in available data often make estimation of small-scale features meaningless. Where covariance patterns can be estimated from the available data, the analysis methods fill gaps, correct sampling errors, and produce spatially and temporally coherent analyzed data sets. As with classical least squares estimation, the reduced space versions also provide theoretical error estimates for analyzed values. The methods are demonstrated on Atlantic monthly sea surface temperature (SST) anomalies for 1856–1991 from the United Kingdom Meteorological Office historical sea surface temperature data set (version MOHSST5). Choice of a reduced space dimension of 30 is shown to be adequate. The analyses are tested by withholding a significant part of the data and prove to be robust and in agreement with their own error estimates; they are also consistent with a partially independent optimal interpolation (OI) analysis by Reynolds and Smith [1994] produced in the National Centers for Environmental Prediction (NCEP)(known as the NCEP OI analysis). A simple statistical model is used to depict the month-to-month SST evolution in the optimal smoother algorithm. Results are somewhat superior to both the Kalman filter, which relies less on the model, and the optimal interpolation, which does not use it at all. The method generalizes a few recent works on using a reduced space for data set analyses. Difficulties of methods which simply fit EOF patterns to observed data are pointed out, and the more complete analysis procedures developed here are suggested as a remedy.

Sensor fusion for mobile robot navigation

Abstract: We review techniques for sensor fusion in robot navigation, emphasizing algorithms for self-location. These find use when the sensor suite of a mobile robot comprises several different sensors, some complementary and some redundant. Integrating the sensor readings, the robot seeks to accomplish tasks such as constructing a map of its environment, locating itself in that map, and recognizing objects that should be avoided or sought. The review describes integration techniques in two categories: low-level fusion is used for direct integration of sensory data, resulting in parameter and state estimates; high-level fusion is used for indirect integration of sensory data in hierarchical architectures, through command arbitration and integration of control signals suggested by different modules. The review provides an arsenal of tools for addressing this (rather ill-posed) problem in machine intelligence, including Kalman filtering, rule-based techniques, behavior based algorithms, and approaches that borrow from information theory, Dempster-Shafer reasoning, fuzzy logic and neural networks.

Estimation and Prediction for a Class of Dynamic Nonlinear Statistical Models

Abstract: A class of nonlinear state-space models, characterized by a single source of randomness, is introduced. A special case, the model underpinning the multiplicative Holt-Winters method of forecasting, is identified. Maximum likelihood estimation based on exponential smoothing instead of a Kalman filter, and with the potential to be applied in contexts involving non-Gaussian disturbances, is considered. A method for computing prediction intervals is proposed and evaluated on both simulated and real data.

Exact Initial Kalman Filtering and Smoothing for Nonstationary Time Series Models

Abstract: This article presents a new exact solution for the initialization of the Kalman filter for state space models with diffuse initial conditions. For example, the regression model with stochastic trend, seasonal and other nonstationary autoregressive integrated moving average components requires a (partially) diffuse initial state vector. The proposed analytical solution is easy to implement and computationally efficient. The exact solution for smoothing is also given. Missing observations are handled in a straightforward manner. All proofs rely on elementary results.

Game theory approach to discrete H/sub /spl infin// filter design

Abstract: A finite-horizon discrete H/sub /spl infin// filter design with a linear quadratic (LQ) game approach is presented. The exogenous inputs composed of the "hostile" noise signals and system initial condition are assumed to be finite energy signals with unknown statistics. The design criterion is to minimize the worst possible amplification of the estimation error signals in terms of the exogenous inputs, which is different from the classical minimum variance estimation error criterion for the modified Wiener or Kalman filter design. The approach can show how far the estimation error can be reduced under an existence condition on the solution to a corresponding Riccati equation. A numerical example is given to compare the performance of the H/sub /spl infin// filter with that of the conventional Kalman filter.

A Framework for Uncertainty and Validation of 3-D RegistrationMethods Based on Points and Frames

Abstract: In this paper, we propose and analyze several methods to estimate a rigid transformation from a set of 3-D matched points or matched frames, which are important features in geometric algorithms. We also develop tools to predict and verify the accuracy of these estimations. The theoretical contributions are: an intrinsic model of noise for transformations based on composition rather than addition; a unified formalism for the estimation of both the rigid transformation and its covariance matrix for points or frames correspondences, and a statistical validation method to verify the error estimation, which applies even when no “ground truth” is available. We analyze and demonstrate on synthetic data that our scheme is well behaved. The practical contribution of the paper is the validation of our transformation estimation method in the case of 3-D medical images, which shows that an accuracy of the registration far below the size of a voxel can be achieved, and in the case of protein substructure matching, where frame features drastically improve both selectivity and complexity.

A jerk model for tracking highly maneuvering targets

Abstract: A model of target motion in three-dimensional space that includes position derivatives up to the third order is developed. Compared with available models, which include terms at the most up to the second derivative, the model introduced in this work, called the jerk model, can more accurately describe agile target maneuvers which are likely to contain significant higher order derivatives. A compatible 4-state Kalman filter to perform tracking in conjunction with the jerk model is also presented, and an initialization procedure for the filter is provided. The improved performance of the jerk model over a lower order model is illustrated through numerical simulation.

Time dependent inversion of geodetic data

Abstract: The recent expansion of permanent Global Positioning System (GPS) networks provides crustal deformation data that are dense in both space and time. While considerable effort has been directed toward using these data for the determination of average crustal velocities, little attention has been given to detecting and estimating transient deformation signals. We introduce here a Network Inversion Filter for estimating the distribution of fault slip in space and time using data from such dense, frequently sampled geodetic networks. Fault slip is expanded in a spatial basis set sk(x) in which the coefficients are time varying, s(x, t) = ∑k=1M ck(t)sk(x) The temporal variation in fault slip is estimated nonparameterically by taking slip accelerations to be random Gaussian increments, so that fault slip is a sum of steady state and integrated random walk components. A state space model for the full geodetic network is formulated, and Kalman filtering methods are used for estimation. Variance parameters, including measurement errors, local benchmark motions, and temporal and spatial smoothing parameters, are estimated by maximum likelihood, which is computed by recursive filtering. Numerical simulations demonstrate that the Network Inversion Filter is capable of imaging fault slip transients, including propagating slip events. The Network Inversion Filter leads naturally to automated methods for detecting anomalous departures from steady state deformation.

Tidal flow forecasting using reduced rank square root filters

Abstract: The Kalman filter algorithm can be used for many data assimilation problems. For large systems, that arise from discretizing partial differential equations, the standard algorithm has huge computational and storage requirements. This makes direct use infeasible for many applications. In addition numerical difficulties may arise if due to finite precision computations or approximations of the error covariance the requirement that the error covariance should be positive semi-definite is violated.

Optimal nonlinear filtering in GPS/INS integration

Abstract: The application of optimal nonlinear/non-Gaussian filtering to the problem of INS/GPS integration in critical situations is described. This approach is made possible by a new technique called particle filtering, and exhibits superior performance when compared with classical suboptimal techniques such as extended Kalman filtering. Particle filtering theory is introduced and GPS/INS integration simulation results are discussed.

Consistent debiased method for converting between polar and Cartesian coordinate systems

Abstract: A significant problem in tracking and estimation is the consistent transformation of uncertain state estimates between Cartesian and spherical coordinate systems. For example, a radar system generates measurements in its own local spherical coordinate system. In order to combine those measurements with those from other radars, however, a tracking system typically transforms all measurements to a common Cartesian coordinate system. The most common approach is to approximate the transformation through linearization. However, this approximation can lead to biases and inconsistencies, especially when the uncertainties on the measurements are large. A number of approaches have been proposed for using higher order transformation modes, but these approaches have found only limited use due to the often enormous implementation burdens incurred by the need to derive Jacobians and Hessians. This paper expands a method for nonlinear propagation which is described in a companion paper. A discrete set of samples are used to capture the first four moments of the untransformed measurement. The transformation is then applied to each of the samples, and the mean and covariance are calculated from the result. It is shown that the performance of the algorithm is comparable to that of fourth order filters, thus ensuring consistency even when the uncertainty is large. It is not necessary to calculate any derivatives, and the algorithm can be extended to incorporate higher order information. The benefits of this algorithm are illustrated in the contexts of autonomous vehicle navigation and missile tracking.

Approximate set-valued observers for nonlinear systems

Abstract: A set-valued observer (SVO) produces a set of possible states based on output measurements and a priori models of exogenous disturbances and noises. Previous work considered linear time-varying systems and unknown-but-bounded exogenous signals. In this case, the sets of possible state vectors take the form of polytopes whose centers are optimal state estimates. These polytopic sets can be computed by solving several small linear programs. An SVO can be constructed conceptually for nonlinear systems; however, the set of possible state vectors no longer takes the form of polytopes, which in turn inhibits their explicit computation. This paper considers an "extended SVO". As in the extended Kalman filter, the state equations are linearized about the state estimate, and a linear SVO is designed along the linearization trajectory. Under appropriate observability assumptions, it is shown that the extended SVO provides an exponentially convergent state estimate in the case of sufficiently small initial condition uncertainty and provides a nondivergent state estimate in the case of sufficiently small exogenous signals.

A smoothly constrained Kalman filter

Abstract: This paper presents the Smoothly Constrained Kalman Filter (SCKF) for nonlinear constraints A constraint is any relation that exists between the state variables Constraints can be treated as perfect observations But, linearization errors can prevent the estimate from converging to the true value Therefore, the SCKF iteratively applies nonlinear constraints as nearly perfect observations, or, equivalently, weakened constraints Integration of new measurements is interlaced with these iterations, which reduces linearization errors and, hence, improves convergence compared to other iterative methods The weakening is achieved by artificially increasing the variance of the nonlinear constraint The paper explains how to choose the weakening values, and when to start and stop the iterative application of the constraint

Modeling and simulation of induction machine vector control with rotor resistance identification

Abstract: This paper shows that it is possible to use available commercial software to model and simulate a vector-controlled induction machine system. The components of a typical vector control system are introduced and methods given for incorporating these in the MATLAB/SIMULINK software package. The identification of rotor resistance is important in vector control, if high-performance torque control is needed, and modeling of the extended Kalman filter (EKF) algorithm for parameter identification is discussed. It is certainly advisable, when feasible, to precede implementation of new algorithms, whether for control or identification purposes, with an extensive simulation phase. Additionally, a technique for generating pulse-width modulation (PWM) phase commands to extend machine operation to higher speeds before field weakening occurs is simulated in a vector-controlled induction machine, driven by a PWM inverter. This demonstrates the versatility of the vector-controlled induction machine system model.

Interval Kalman filtering

Abstract: The classical Kalman filtering technique is extended to interval linear systems with the same statistical assumptions on noise, for which the classical technique is no longer applicable. Necessary interval analysis, particularly the notion of interval expectation, is reviewed and introduced. The interval Kalman filter (IKF) is then derived, which has the same structure as the classical algorithm, using no additional analysis or computation from such as H/sup /spl infin//-mathematics. A suboptimal IKF is suggested next, for the purpose of real-time implementation. Finally, computer simulations are shown to compare the new interval Kalman filtering algorithm with the classical Kalman filtering scheme and some other existing robust Kalman filtering methods.