Classical Extended Kalman Filter Research Articles

A perturbation method for estimation of dynamic systems

A novel framework called the Perturbed Jth Moment Extended Kalman Filter (PJMEKF), based on a classical perturbation technique is proposed for estimating the states of a nonlinear dynamical system from sensor measurements. This method falls under a class of architectures under investigation primarily to study the interplay of major issues in nonlinear estimation such as nonlinearity, measurement sparsity, and initial condition uncertainty in an environment with low levels of process noise. Taylor series expansion of the departure motion dynamics about the best estimate is used to derive a series representation of the unforced motion. It is found that such series representation evolves as a set of differential equations that force each other in a cascade manner, adding up to give the unforced motion (in a so-called “triangular” structure). This formal perturbation solution for the departure motion dynamics is used in deriving the differential equations governing the time evolution of the high order statistical moments of the estimation error. These tensor differential equations are found to possess a similar high order triangular structure in addition to being symmetric (in N tensorial dimensions and we appropriately term the evolution equations as Tensor Lyapunov Equations of statistical moment perturbations). Elegance of the tensor differential equations thus derived is accompanied by the computational advantages due to symmetry in all tensorial dimensions. A vector matrix representation of tensors is proposed with which the representation and solution of the tensor differential equations can be carried out effectively. Approximations are introduced to incorporate low levels of process noise forcing function in the propagation phase of the moment equations. The statistics thus propagated are used in a filtering framework to estimate the state vector of a nonlinear system from noisy measurements, within the traditional Kalman update paradigm. The Kalman gain thus determined is utilized in updating all high order moments in preparation for the subsequent propagation phase leading to improved estimation accuracy. The filter developed is applied to an orbit estimation problem and comparisons are presented with classical extended Kalman filter.

A high order method for estimation of dynamic systems

An analytical approach is presented for developing an estimation framework (called the Jth Moment Extended Kalman Filter (JMEKF)). This forms an important component of a class of architectures under investigation to study the interplay of major issues in nonlinear estimation such as model nonlinearity, measurement sparsity and initial condition uncertainty in the presence of low process noise. Utilizing an automated nonlinear expansion of the model about the current best estimated trajectory, a Jth order approximate solution for the departure motion dynamics about a nominal trajectory is derived in the form of state transition tensors. This solution is utilized in evaluating the evolution of statistics of the departure motion as a function of the statistics of initial conditions. The statistics thus obtained are used in the determination of a state estimate assuming a Kalman update structure. Central to the state transition tensor integration about a nominal trajectory, is the high order sensitivity calculations of the nonlinear models (dynamics and measurement), being automated by OCEA (Object Oriented Coordinate Embedding Method), a computational tool generating the required various order partials of the system differential equations without user intervention. Working in tandem with an OCEA automation of the derivation of the state transition tensor differential equations is a vector matrix representation structure of tensors of arbitrary rank, facilitating faster and more accurate computations. High order moment update equations are derived to incorporate the statistical effects of the innovations process more rigorously, improving the effectiveness of the estimation scheme. Numerical simulations on an orbit estimation example investigate the gain obtained in using the proposed methodology in situations where the classical extended Kalman filter’s domain of convergence is smaller. The orbit estimation example presented examines a situation that requires us to determine the position and velocity state of the orbiter from range, azimuth and elevation measurements being made available sparsely.

Model error and sequential data assimilation: A deterministic formulation

AbstractData assimilation schemes are confronted with the presence of model errors arising from the imperfect description of atmospheric dynamics. These errors are usually modelled on the basis of simple assumptions such as bias, white noise, and first‐order Markov process. In the present work, a formulation of the sequential extended Kalman filter is proposed, based on recent findings on the universal deterministic behaviour of model errors in marked contrast with previous approaches. This new scheme is applied in the context of a spatially distributed system proposed by Lorenz. First, it is found that, for short times, the estimation error is accurately approximated by an evolution law in which the variance of the model error (assumed to be a deterministic process) evolves according to a quadratic law, in agreement with the theory. Moreover, the correlation with the initial condition error appears to play a secondary role in the short‐time dynamics of the estimation error covariance. Second, the deterministic description of the model error evolution, incorporated into the classical extended Kalman filter equations, reveals that substantial improvements of the filter accuracy can be gained compared with the classical white‐noise assumption. The universal short‐time quadratic law for the evolution of the model error covariance matrix seems very promising for modelling estimation error dynamics in sequential data assimilation. Copyright © 2008 Royal Meteorological Society

A local geopotential model for implementation of underwater passive navigation

A main aspect of underwater passive navigation is how to identify the vehicle location on an existing gravity map, and several matching algorithms as ICCP and SITAN are the most prevalent methods that many scholars are using. In this paper, a novel algorithm that is different from matching algorithms for passive navigation is developed. The algorithm implements underwater passive navigation by directly estimating the inertial errors through Kalman filter algorithm, and the key part of this implementation is a Fourier series-based local geopotential model. Firstly, the principle of local geopotential model based on Fourier series is introduced in this paper, thus the discrete gravity anomalies data can be expressed analytically with respect to geographic coordinates to establish the observation equation required in the application of Kalman filter. Whereafter, the indicated gravity anomalies can be gotten by substituting the inertial positions to existing gravity anomalies map. Finally, the classical extended Kalman filter is introduced with the differences between measured gravity and indicated gravity used as observations to optimally estimate the errors of the inertial navigation system (INS). This navigation algorithm is tested on simulated data with encouraging results. Although this algorithm is developed for underwater navigation using gravity data, it is equally applicable to other domains, for example vehicle navigation on magnetic or terrain data.

Centralized and decentralized process and sensor fault monitoring using data fusion based on adaptive extended Kalman filter algorithm

This paper presents an integrated design framework to utilize multi-sensor data fusion (MSDF) techniques for process monitoring enhancement to detect and diagnose sensor and process faults. Two different distributed and centralized architectures are presented to integrate the multi-sensor data based on extended Kalman filter (EKF) data fusion algorithm. The distributed integration architecture uses the state-vector fusion method, while the centralized integration architecture is based on the output augmented fusion (OAF) method. The usual approach in the classical EKF implementation is based on the assumption of constant diagonal matrices for both the process and measurement covariances. This inflexible constant covariance set-up may cause degradation in the EKF performance. A new adaptive modified EKF (AMEKF) algorithm has been developed to prevent the filter divergence and hence leading to an improved EKF estimation. A set of simulation studies have been conducted to demonstrate the performances of the proposed adaptive and non-adaptive process monitoring approaches on a continuous stirred tank reactor (CSTR) benchmark problem. The sensor fault studies include the sensor faults due to drift in calibration and drift in sensor degradation anomalies. Whereas, the process faults consist of four probable CSTR faults in cascaded single, double, triple and quadruple set-up.

Hybrid three-dimensional camera pose estimation using particle filter sensor fusion

To properly align objects in the real and virtual worlds in an augmented reality (AR) space it is essential to keep tracking the camera's exact three-dimensional position and orientation (camera pose). State-of-the-art analysis shows that traditional vision-based or inertial sensor-based solutions are not adequate when used individually. Sensor fusion for hybrid tracking has become an active research direction during the past few years, although how to do it in a robust and principled way is still an open problem. In this paper, we develop a hybrid camera pose-tracking system that combines vision and inertial sensor technologies. We propose to use the particle filter framework for the sensor fusion system. Particle filters are sequential Monte-Carlo methods based upon a point mass (or 'particle') representation of probability densities, which can be applied to any state space model and which generalize the traditional Kalman filtering methods. We have tested our algorithm to evaluate its performance and have compared the results obtained by the particle filter with those given by a classical extended Kalman filter. Experimental results are presented

Adaptive Estimation of FCG Using Nonlinear State-Space Models

In this paper, an efficient adaptive nonlinear algorithm for estimation and identification, the so-called adaptive Lainiotis filter (ALF), is applied to the problem of fatigue crack growth (FCG) estimation, identification, and prediction of the final crack (failure). A suitable nonlinear state-space FCG model is introduced for both ALF and extended Kalman filter (EKF). Both algorithms are tested in order to compare their efficiency. Through extensive analysis and simulation, it is demonstrated that the ALF has superior performance both in FCG estimation, as well as in predicting the remaining lifetime to failure. Furthermore, it is shown that the ALF is faster and easier to implement in a parallel/distributed processing mode, and much more robust than the classic EKF.

Limits to the consistency of EKF-based SLAM

This paper analyzes the consistency of the classical extended Kalman filter (EKF) solution to the simultaneous localization and map building (SLAM) problem. Our results show that in large environments the map quickly becomes inconsistent due to linearization errors. We propose a new EKF-based SLAM algorithm,robocentric-mapping, that greatly reduces linearization errors, improving map consistency. We also present results showing that large-scale mapping methods based on building local maps with a local uncertainty representation (Tardós et al.,2002) have better consistency than methods that work with global uncertainties.

Nonlinear Statistical Approach for Aeroelastic Response Prediction

Aging aircraft and combat aircraft that carry heavy external stores potentially face problems arising from nonlinearities in structure. An expert data mining system is proposed that is capable of predicting the asymptotic behavior of an aeroelastic system with structural nonlinearities represented by polynomial restoring forces or freeplay models. The input is represented only by a limited set of transient data. The output provides a long-term nonlinear aeroelastic response, and the prediction is made when certain rule-based reasoning conditions are satisfied. An attractive feature of this new approach is that no information about the system parameters is needed. In the prediction module, we propose two methods, based on nonlinear time series models and the unscented Kalman filter. To our knowledge, these approaches have not been reported so far for predicting the long-term nonlinear aeroelastic responses. Compared with the classical extended Kalman filter, the unscented filter does not require differentiability and can be applied to nonlinear aeroelastic models with freeplay and hysteresis. The performances of the expert data mining system are demonstrated for simulated data and wind-tunnel experimental aeroelastic data resulting from a two-degree-of-freedom airfoil oscillating in pitch and plunge.

Triangulation-based fusion of sonar data with application in robot pose tracking

In this paper a sensor fusion scheme, called triangulation-based fusion (TBF) of sonar data, is presented. This algorithm delivers stable natural point landmarks, which appear in practically all indoor environments, i.e., vertical edges like door posts, table legs, and so forth. The landmark precision is in most cases within centimeters. The TBF algorithm is implemented as a voting scheme, which groups sonar measurements that are likely to have hit the same object in the environment. The algorithm has low complexity and is sufficiently fast for most mobile robot applications. As a case study, we apply the TBF algorithm to robot pose tracking. The pose tracker is implemented as a classic extended Kalman filter, which use odometry readings for the prediction step and TBF data for measurement updates. The TBF data is matched to pre-recorded reference maps of landmarks in order to measure the robot pose. In corridors, complementary TBF data measurements from the walls are used to improve the orientation and position estimate. Experiments demonstrate that the pose tracker is robust enough for handling kilometer distances in a large scale indoor environment containing a sufficiently dense landmark set.

Efficient algorithms of clustering adaptive nonlinear filters

This paper proposes a new class of efficient adaptive nonlinear filters whose estimation error performance (in a minimum mean square sense) is superior to that of competing approximate nonlinear filters, e.g., the well-known extended Kalman filter (EKF). The proposed filters include as special cases both the EKF and previously proposed partitioning filters. The new methodology performs an adaptive selection of appropriate reference points for linearization from an ensemble of generated trajectories that have been processed and clustered accordingly to span the whole state space of the desired signal. Through a series of simulation examples, the approach is shown significantly superior to the classical EKF with comparable computational burden.

Tracking for augmented reality on wearable computers

Wearable computers afford a degree of mobility that makes tracking for augmented reality difficult. This paper presents a novel object-centric tracking architecture for presenting augmented reality media in spatial relationships to objects, regardless of the objects' positions or motions in the world. The advance this system provides is the ability to sense and integrate new features into its tracking database, thereby extending the tracking region automatically. A “lazy evaluation” of the structure from motion problem uses images obtained from a single calibrated moving camera and applies recursive filtering to identify and estimate the 3D positions of new features. We evaluate the performance of two filters; a classic Extended Kalman Filter (EKF) and a filter based on a Recursive-Average of Covariances (RAC). Some implementation issues and results are discussed in conclusion.

A new class of efficient adaptive nonlinear filters (ANLF)

A multilinearization procedure is described, with the use of which a new class of algorithms for nonlinear filtering can be realized. The methodology targets on adaptively selecting the best reference points for linearization from an ensemble of generated trajectories that span the whole state space of the desired signal. Through simulations, the approach is shown to be significantly superior to the classical extended Kalman filter and comparable in computational burden.

A partitioned adaptive approach to nonlinear channel equalization

The problem of identifying digital transmitted symbols over nonlinear communication channels is addressed. The equalization scenario is considered from the decision point of view, and constitutes a joint identification and estimation situation due to incomplete knowledge of the system model. A new class of multilinearization algorithms for nonlinear systems is derived according to partitioning theory concepts. The procedure targets on adaptively selecting the best reference points for linearization from an ensemble of generated trajectories that span the whole state space of the desired signal. In the various simulations examined, the partitioned-based equalizer is found superior to the classical extended Kalman filter.

Tracking an incoming ballistic missile using an extended interval Kalman filter

The important tracking problem by radar of an incoming ballistic missile system, which contains uncertainty in modeling and noise in both dynamics and measurements, is studied. The classical extended Kalman filter (EKF) is no longer applicable to such an uncertain system, and so a new extended interval Kalman filter (EIKF) is developed for tracking the missile system. Computer simulation is presented to show the effectiveness of the EIKF algorithm for this uncertain and nonlinear ballistic missile tracking problem.