Showing papers on "Kalman filter published in 2012"

Introduction to Random Signals and Applied Kalman Filtering with Matlab Exercises

Abstract: PART 1: RANDOM SIGNALS BACKGROUND Chapter 1 Probability and Random Variables: A Review Chapter 2 Mathematical Description of Random Signals Chapter 3 Linear Systems Response, State-space Modeling and Monte Carlo Simulation PART 2: KALMAN FILTERING AND APPLICATIONS Chapter 4 Discrete Kalman Filter Basics Chapter 5 Intermediate Topics on Kalman Filtering Chapter 6 Smoothing and Further Intermediate Topics Chapter 7 Linearization, Nonlinear Filtering and Sampling Bayesian Filters Chapter 8 the "Go-Free" Concept, Complementary Filter and Aided Inertial Examples Chapter 9 Kalman Filter Applications to the GPS and Other Navigation Systems APPENDIX A. Laplace and Fourier Transforms APPENDIX B. The Continuous Kalman Filter

Kalman Filter for Robot Vision: A Survey

Abstract: Kalman filters have received much attention with the increasing demands for robotic automation. This paper briefly surveys the recent developments for robot vision. Among many factors that affect the performance of a robotic system, Kalman filters have made great contributions to vision perception. Kalman filters solve uncertainties in robot localization, navigation, following, tracking, motion control, estimation and prediction, visual servoing and manipulation, and structure reconstruction from a sequence of images. In the 50th anniversary, we have noticed that more than 20 kinds of Kalman filters have been developed so far. These include extended Kalman filters and unscented Kalman filters. In the last 30 years, about 800 publications have reported the capability of these filters in solving robot vision problems. Such problems encompass a rather wide application area, such as object modeling, robot control, target tracking, surveillance, search, recognition, and assembly, as well as robotic manipulation, localization, mapping, navigation, and exploration. These reports are summarized in this review to enable easy referral to suitable methods for practical solutions. Representative contributions and future research trends are also addressed in an abstract level.

Accurate Pedestrian Indoor Navigation by Tightly Coupling Foot-Mounted IMU and RFID Measurements

Abstract: We present a new method to accurately locate persons indoors by fusing inertial navigation system (INS) techniques with active RFID technology. A foot-mounted inertial measuring units (IMUs)-based position estimation method, is aided by the received signal strengths (RSSs) obtained from several active RFID tags placed at known locations in a building. In contrast to other authors that integrate IMUs and RSS with a loose Kalman filter (KF)-based coupling (by using the residuals of inertial- and RSS-calculated positions), we present a tight KF-based INS/RFID integration, using the residuals between the INS-predicted reader-to-tag ranges and the ranges derived from a generic RSS path-loss model. Our approach also includes other drift reduction methods such as zero velocity updates (ZUPTs) at foot stance detections, zero angular-rate updates (ZARUs) when the user is motionless, and heading corrections using magnetometers. A complementary extended Kalman filter (EKF), throughout its 15-element error state vector, compensates the position, velocity and attitude errors of the INS solution, as well as IMU biases. This methodology is valid for any kind of motion (forward, lateral or backward walk, at different speeds), and does not require an offline calibration for the user gait. The integrated INS+RFID methodology eliminates the typical drift of IMU-alone solutions (approximately 1% of the total traveled distance), resulting in typical positioning errors along the walking path (no matter its length) of approximately 1.5 m.

Understanding the Basis of the Kalman Filter Via a Simple and Intuitive Derivation

Abstract: T his article provides a simple and intuitive derivation of the Kalman filter, with the aim of teaching this useful tool to students from disciplines that do not require a strong mathematical background. The most complicated level of mathematics required to understand this derivation is the ability to multiply two Gaussian functions together and reduce the result to a compact form. The Kalman filter is over 50 years old but is still one of the most important and common data fusion algorithms in use today. Named after Rudolf E. Kalman, the great success of the Kalman filter is due to its small computational requirement, elegant recursive properties, and its status as the optimal estimator for one-dimensional linear systems with Gaussian error statistics [1] . Typical uses of the Kalman filter include smoothing noisy data and providing estimates of parameters of interest. Applications include global positioning system receivers, phaselocked loops in radio equipment, smoothing the output from laptop trackpads, and many more. From a theoretical standpoint, the Kalman filter is an algorithm permitting exact inference in a linear dynamical system, which is a Bayesian model similar to a hidden Markov model but where the state space of the latent variables is continuous and where all latent and observed variables have a Gaussian distribution (often a multivariate Gaussian distribution). The aim of this lecture note is to permit people who find this description confusing or terrifying to understand the basis of the Kalman filter via a simple and intuitive derivation.

Camera-based navigation of a low-cost quadrocopter

Abstract: In this paper, we describe a system that enables a low-cost quadrocopter coupled with a ground-based laptop to navigate autonomously in previously unknown and GPS-denied environments. Our system consists of three components: a monocular SLAM system, an extended Kalman filter for data fusion and state estimation and a PID controller to generate steering commands. Next to a working system, the main contribution of this paper is a novel, closed-form solution to estimate the absolute scale of the generated visual map from inertial and altitude measurements. In an extensive set of experiments, we demonstrate that our system is able to navigate in previously unknown environments at absolute scale without requiring artificial markers or external sensors. Furthermore, we show (1) its robustness to temporary loss of visual tracking and significant delays in the communication process, (2) the elimination of odometry drift as a result of the visual SLAM system and (3) accurate, scale-aware pose estimation and navigation.

Some Relations Between Extended and Unscented Kalman Filters

Abstract: The unscented Kalman filter (UKF) has become a popular alternative to the extended Kalman filter (EKF) during the last decade. UKF propagates the so called sigma points by function evaluations using the unscented transformation (UT), and this is at first glance very different from the standard EKF algorithm which is based on a linearized model. The claimed advantages with UKF are that it propagates the first two moments of the posterior distribution and that it does not require gradients of the system model. We point out several less known links between EKF and UKF in terms of two conceptually different implementations of the Kalman filter: the standard one based on the discrete Riccati equation, and one based on a formula on conditional expectations that does not involve an explicit Riccati equation. First, it is shown that the sigma point function evaluations can be used in the classical EKF rather than an explicitly linearized model. Second, a less cited version of the EKF based on a second-order Taylor expansion is shown to be quite closely related to UKF. The different algorithms and results are illustrated with examples inspired by core observation models in target tracking and sensor network applications.

Understanding the Basis of the Kalman Filter Via a Simple and Intuitive Derivation [Lecture Notes]

Abstract: This article provides a simple and intuitive derivation of the Kalman filter, with the aim of teaching this useful tool to students from disciplines that do not require a strong mathematical background. The most complicated level of mathematics required to understand this derivation is the ability to multiply two Gaussian functions together and reduce the result to a compact form.

Coupled Distributed Estimation and Control for Mobile Sensor Networks

Abstract: In this paper, we introduce a theoretical framework for coupled distributed estimation and motion control of mobile sensor networks for collaborative target tracking. We use a Fisher Information theoretic metric for quality of sensed data. The mobile sensing agents seek to improve the information value of their sensed data while maintaining a safe-distance from other neighboring agents (i.e., perform information-driven flocking). We provide a formal stability analysis of continuous Kalman-Consensus filtering (KCF) algorithm on a mobile sensor network with a flocking-based mobility control model. The discrete-time counterpart of this coupled estimation and control algorithm is successfully applied to tracking of two types of targets with stochastic linear and nonlinear dynamics.

Interacting Multiple Model Filter-Based Sensor Fusion of GPS With In-Vehicle Sensors for Real-Time Vehicle Positioning

Abstract: Vehicle position estimation for intelligent vehicles requires not only highly accurate position information but reliable and continuous information provision as well. A low-cost Global Positioning System (GPS) receiver has widely been used for conventional automotive applications, but it does not guarantee accuracy, reliability, or continuity of position data when GPS errors occur. To mitigate GPS errors, numerous Bayesian filters based on sensor fusion algorithms have been studied. The estimation performance of Bayesian filters primarily relies on the choice of process model. For this reason, the change in vehicle dynamics with driving conditions should be addressed in the process model of the Bayesian filters. This paper presents a positioning algorithm based on an interacting multiple model (IMM) filter that integrates low-cost GPS and in-vehicle sensors to adapt the vehicle model to various driving conditions. The model set of the IMM filter is composed of a kinematic vehicle model and a dynamic vehicle model. The algorithm developed in this paper is verified via intensive simulation and evaluated through experimentation with a real-time embedded system. Experimental results show that the performance of the positioning system is accurate and reliable under a wide range of driving conditions.

An Alternative Method for Power System Dynamic State Estimation Based on Unscented Transform

Abstract: An efficient, timely, and accurate state estimation is a prerequisite for most energy management system (EMS) applications in power system control centers. The emerging wide-area measurement systems (WAMSs) offer new opportunities for developing more effective methods to monitor power system dynamics online. Recently, alternative methods for power system state estimation have caught much attention. Due to the nonlinearity of state transition and observation equation, linearization and Jacobian matrix calculation are indispensible in the existing methods of power system state estimation. This makes WAMS' high performance compromised by burdensome calculation. In order to overcome the drawbacks, this study tries to develop an effective state estimation method without the linearization and Jacobian matrix calculation. Firstly, unscented transformation is introduced as an effective method to calculate the means and covariances of a random vector undergoing a nonlinear transformation. Secondly, by embedding the unscented transformation into the Kalman filter process, a method is developed for power system dynamic state estimation. Finally, some simulation results are presented showing accuracy and easier implementation of the new method.

The Ensemble Kalman Filter for Inverse Problems

Abstract: The Ensemble Kalman filter (EnKF) was introduced by Evensen in 1994 [10] as a novel method for data assimilation: state estimation for noisily observed time-dependent problems. Since that time it has had enormous impact in many application domains because of its robustness and ease of implementation, and numerical evidence of its accuracy. In this paper we propose the application of an iterative ensemble Kalman method for the solution of a wide class of inverse problems. In this context we show that the estimate of the unknown function that we obtain with the ensemble Kalman method lies in a subspace A spanned by the initial ensemble. Hence the resulting error may be bounded above by the error found from the best approximation in this subspace. We provide numerical experiments which compare the error incurred by the ensemble Kalman method for inverse problems with the error of the best approximation in A, and with variants on traditional least-squares approaches, restricted to the subspace A. In so doing we demonstrate that the ensemble Kalman method for inverse problems provides a derivative-free optimization method with comparable accuracy to that achieved by traditional least-squares approaches. Furthermore, we also demonstrate that the accuracy is of the same order of magnitude as that achieved by the best approximation. Three examples are used to demonstrate these assertions: inversion of a compact linear operator; inversion of piezometric head to determine hydraulic conductivity in a Darcy model of groundwater flow; and inversion of Eulerian velocity measurements at positive times to determine the initial condition in an incompressible fluid.

Filtering Complex Turbulent Systems

Abstract: Preface 1. Introduction and overview: mathematical strategies for filtering turbulent systems Part I. Fundamentals: 2. Filtering a stochastic complex scalar: the prototype test problem 3. The Kalman filter for vector systems: reduced filters and a three-dimensional toy model 4. Continuous and discrete Fourier series and numerical discretization Part II. Mathematical Guidelines for Filtering Turbulent Signals: 5. Stochastic models for turbulence 6. Filtering turbulent signals: plentiful observations 7. Filtering turbulent signals: regularly spaced sparse observations 8. Filtering linear stochastic PDE models with instability and model error Part III. Filtering Turbulent Nonlinear Dynamical Systems: 9. Strategies for filtering nonlinear systems 10. Filtering prototype nonlinear slow-fast systems 11. Filtering turbulent nonlinear dynamical systems by finite ensemble methods 12. Filtering turbulent nonlinear dynamical systems by linear stochastic models 13. Stochastic parameterized extended Kalman filter for filtering turbulent signal with model error 14. Filtering turbulent tracers from partial observations: an exactly solvable test model 15. The search for efficient skilful particle filters for high dimensional turbulent dynamical systems References Index.

Shoulder and Elbow Joint Angle Tracking With Inertial Sensors

Abstract: Wearable inertial systems have recently been used to track human movement in and outside of the laboratory. Continuous monitoring of human movement can provide valuable information relevant to individuals’ level of physical activity and functional ability. Traditionally, orientation has been calculated by integrating the angular velocity from gyroscopes. However, a small drift in the measured velocity leads to increasing integration error over time. To compensate that drift, complementary data from accelerometers are normally fused into tracking systems using the Kalman or extended Kalman filter. In this study, we combine kinematic models designed for control of robotic arms with state-space methods to continuously estimate the angles of human shoulder and elbow using two wearable inertial measurement units. We use the unscented Kalman filter to implement the nonlinear state-space inertial tracker. Shoulder and elbow joint angles obtained from 8 subjects using our inertial tracker were compared to the angles obtained from an optical-tracking reference system. On average, there was an RMS angle error of less than 8 $^\circ$ for all shoulder and elbow angles. The average correlation coefficient for all movement tasks among all subjects was $r\ge \hbox{0.95}$ . This agreement between our inertial tracker and the optical reference system was obtained for both regular and fast-speed movement of the arm. The same method can be used to track movement of other joints.

Estimation of Dynamic Term Structure Models

Abstract: We study the finite-sample properties of some of the standard techniques used to estimate modern term structure models. For sample sizes and models similar to those used in most empirical work, we reach three surprising conclusions. First, while maximum likelihood works well for simple models, it produces strongly biased parameter estimates when the model includes a flexible specification of the dynamics of interest rate risk. Second, despite having the same asymptotic efficiency as maximum likelihood, the small-sample performance of Efficient Method of Moments (a commonly used method for estimating complicated models) is unacceptable even in the simplest term structure settings. Third, the linearized Kalman filter is a tractable and reasonably accurate estimation technique, which we recommend in settings where maximum likelihood is impractical.

Diffusion Kalman Filtering Based on Covariance Intersection

Abstract: This paper is concerned with distributed Kalman filtering for linear time-varying systems over multiagent sensor networks. We propose a diffusion Kalman filtering algorithm based on the covariance intersection method, where local estimates are fused by incorporating the covariance information of local Kalman filters. Our algorithm leads to a stable estimate for each agent regardless of whether the system is uniformly observable locally by the measurements of its neighbors which include the measurements of itself as long as the system is uniformly observable by the measurements of all the agents and the communication is sufficiently fast compared to the sampling. Simulation results validate the effectiveness of the proposed distributed Kalman filtering algorithm.

Approximate Inference in State-Space Models With Heavy-Tailed Noise

Abstract: State-space models have been successfully applied across a wide range of problems ranging from system control to target tracking and autonomous navigation. Their ubiquity stems from their modeling flexibility, as well as the development of a battery of powerful algorithms for estimating the state variables. For multivariate models, the Gaussian noise assumption is predominant due its convenient computational properties. In some cases, anyhow, this assumption breaks down and no longer holds. We propose a novel approach to extending the applicability of this class of models to a wider range of noise distributions without losing the computational advantages of the associated algorithms. The estimation methods we develop parallel the Kalman filter and thus are readily implemented and inherit the same order of complexity. We derive all of the equations and algorithms from first principles. In order to validate the performance of our approach, we present specific instances of non-Gaussian state-space models and test their performance on experiments with synthetic and real data.

Event-based state estimation with variance-based triggering

Abstract: An event-based state estimation scenario is considered where a sensor sporadically transmits observations of a scalar linear process to a remote estimator. The remote estimator is a time-varying Kalman filter. The triggering decision is based on the estimation variance: the sensor runs a copy of the remote estimator and transmits a measurement if the associated measurement prediction variance exceeds a tolerable threshold. The resulting variance iteration is a new type of Riccati equation with switching that corresponds to the availability or unavailability of a measurement and depends on the variance at the previous step. We study asymptotic properties of the variance iteration and, in particular, asymptotic convergence to a periodic solution.

UAV Attitude Estimation Using Unscented Kalman Filter and TRIAD

Abstract: A main problem in autonomous vehicles in general, and in unmanned aerial vehicles (UAVs) in particular, is the determination of the attitude angles. A novel method to estimate these angles using off-the-shelf components is presented. This paper introduces an attitude heading reference system (AHRS) based on the unscented Kalman filter (UKF) using the three-axis attitude determination (TRIAD) algorithm as the observation model. The performance of the method is assessed through simulations and compared to an AHRS based on the extended Kalman filter (EKF). The paper presents field experiment results using a real fixed-wing UAV. The results show good real-time performance with low computational cost in a microcontroller.

Real-Time Lagrangian Traffic State Estimator for Freeways

Abstract: Freeway traffic state estimation and prediction are central components in real-time traffic management and information applications. Model-based traffic state estimators consist of a dynamic model for the state variables (e.g., a first- or second-order macroscopic traffic flow model), a set of observation equations relating sensor observations to the system state (e.g., the fundamental diagrams), and a data-assimilation technique to combine the model predictions with the sensor observations [e.g., the extended Kalman filter (EKF)]. Commonly, both process and observation models are formulated in Eulerian (space-time) coordinates. Recent studies have shown that this model can be formulated and solved more efficiently and accurately in Lagrangian (vehicle number-time) coordinates. In this paper, we propose a new model-based state estimator based on the EKF technique, in which the discretized Lagrangian Lighthill-Whitham and Richards (LWR) model is used as the process equation, and in which observation models for both Eulerian and Lagrangian sensor data (from loop detectors and vehicle trajectories, respectively) are incorporated. This Lagrangian state estimator is validated and compared with a Eulerian state estimator based on the same LWR model using an empirical microscopic traffic data set from the U.K. The results indicate that the Lagrangian estimator is significantly more accurate and offers computational and theoretical benefits over the Eulerian approach.

Target Tracking in Wireless Sensor Networks Based on the Combination of KF and MLE Using Distance Measurements

Abstract: A common technical difficulty in target tracking in a wireless sensor network is that individual homogeneous sensors only measure their distances to the target whereas the state of the target composes of its position and velocity in the Cartesian coordinates. That is, the senor measurements are nonlinear in the target state. Extended Kalman filtering is a commonly used method to deal with the nonlinearity, but this often leads to unsatisfactory or even unstable tracking performances. In this paper, we present a new target tracking approach which avoids the instability problem and offers superior tracking performances. We first propose an improved noise model which incorporates both additive noises and multiplicative noises in distance sensing. We then use a maximum likelihood estimator for prelocalization to remove the sensing nonlinearity before applying a standard Kalman filter. The advantages of the proposed approach are demonstrated via experimental and simulation results.

Visual tracking of human visitors under variable-lighting conditions for a responsive audio art installation

Abstract: For a responsive audio art installation in a skylit atrium, we introduce a single-camera statistical segmentation and tracking algorithm. The algorithm combines statistical background image estimation, per-pixel Bayesian segmentation, and an approximate solution to the multi-target tracking problem using a bank of Kalman filters and Gale-Shapley matching. A heuristic confidence model enables selective filtering of tracks based on dynamic data. We demonstrate that our algorithm has improved recall and F 2 -score over existing methods in OpenCV 2.1 in a variety of situations. We further demonstrate that feedback between the tracking and the segmentation systems improves recall and F 2 -score. The system described operated effectively for 5–8 hours per day for 4 months; algorithms are evaluated on video from the camera installed in the atrium. Source code and sample data is open source and available in OpenCV.

Parameter estimation by ensemble Kalman filters with transformed data: Approach and application to hydraulic tomography

Abstract: [1] Ensemble Kalman filters (EnKFs) are a successful tool for estimating state variables in atmospheric and oceanic sciences. Recent research has prepared the EnKF for parameter estimation in groundwater applications. EnKFs are optimal in the sense of Bayesian updating only if all involved variables are multivariate Gaussian. Subsurface flow and transport state variables, however, generally do not show Gaussian dependence on hydraulic log conductivity and among each other, even if log conductivity is multi-Gaussian. To improve EnKFs in this context, we apply nonlinear, monotonic transformations to the observed states, rendering them Gaussian (Gaussian anamorphosis, GA). Similar ideas have recently been presented by Beal et al. (2010) in the context of state estimation. Our work transfers and adapts this methodology to parameter estimation. Additionally, we address the treatment of measurement errors in the transformation and provide several multivariate analysis tools to evaluate the expected usefulness of GA beforehand. For illustration, we present a first-time application of an EnKF to parameter estimation from 3-D hydraulic tomography in multi-Gaussian log conductivity fields. Results show that (1) GA achieves an implicit pseudolinearization of drawdown data as a function of log conductivity and (2) this makes both parameter identification and prediction of flow and transport more accurate. Combining EnKFs with GA yields a computationally efficient tool for nonlinear inversion of data with improved accuracy. This is an attractive benefit, given that linearization-free methods such as particle filters are computationally extremely demanding.

Improving the accuracy of EKF-based visual-inertial odometry

Abstract: In this paper, we perform a rigorous analysis of EKF-based visual-inertial odometry (VIO) and present a method for improving its performance. Specifically, we examine the properties of EKF-based VIO, and show that the standard way of computing Jacobians in the filter inevitably causes inconsistency and loss of accuracy. This result is derived based on an observability analysis of the EKF's linearized system model, which proves that the yaw erroneously appears to be observable. In order to address this problem, we propose modifications to the multi-state constraint Kalman filter (MSCKF) algorithm [1], which ensure the correct observability properties without incurring additional computational cost. Extensive simulation tests and real-world experiments demonstrate that the modified MSCKF algorithm outperforms competing methods, both in terms of consistency and accuracy.

Estimation of Attitude and External Acceleration Using Inertial Sensor Measurement During Various Dynamic Conditions

Abstract: This paper proposes a Kalman filter-based attitude (ie, roll and pitch) estimation algorithm using an inertial sensor composed of a triaxial accelerometer and a triaxial gyroscope In particular, the proposed algorithm has been developed for accurate attitude estimation during dynamic conditions, in which external acceleration is present Although external acceleration is the main source of the attitude estimation error and despite the need for its accurate estimation in many applications, this problem that can be critical for the attitude estimation has not been addressed explicitly in the literature Accordingly, this paper addresses the combined estimation problem of the attitude and external acceleration Experimental tests were conducted to verify the performance of the proposed algorithm in various dynamic condition settings and to provide further insight into the variations in the estimation accuracy Furthermore, two different approaches for dealing with the estimation problem during dynamic conditions were compared, ie, threshold-based switching approach versus acceleration model-based approach Based on an external acceleration model, the proposed algorithm was capable of estimating accurate attitudes and external accelerations for short accelerated periods, showing its high effectiveness during short-term fast dynamic conditions Contrariwise, when the testing condition involved prolonged high external accelerations, the proposed algorithm exhibited gradually increasing errors However, as soon as the condition returned to static or quasi-static conditions, the algorithm was able to stabilize the estimation error, regaining its high estimation accuracy