Invariant extended Kalman filter

About: Invariant extended Kalman filter is a research topic. Over the lifetime, 7079 publications have been published within this topic receiving 187702 citations.

On the identification of variances and adaptive Kalman filtering

Abstract: A Kalman filter requires an exact knowledge of the process noise covariance matrix Q and the measurement noise covariance matrix R . Here we consider the case in which the true values of Q and R are unknown. The system is assumed to be constant, and the random inputs are stationary. First, a correlation test is given which checks whether a particular Kalman filter is working optimally or not. If the filter is suboptimal, a technique is given to obtain asymptotically normal, unbiased, and consistent estimates of Q and R . This technique works only for the case in which the form of Q is known and the number of unknown elements in Q is less than n \times r where n is the dimension of the state vector and r is the dimension of the measurement vector. For other cases, the optimal steady-state gain K op is obtained directly by an iterative procedure without identifying Q . As a corollary, it is shown that the steady-state optimal Kalman filter gain K op depends only on n \times r linear functionals of Q . The results are first derived for discrete systems. They are then extended to continuous systems. A numerical example is given to show the usefulness of the approach.

Kalman Filtering: Theory and Practice

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

Kalman Filtering for Spacecraft Attitude Estimation

Abstract: HIS report reviews the methods of Kalman filtering in attitude estimation and their development over the last two decades. This review is not intended to be complete but is limited to algorithms suitable for spacecraft equipped with three-axis gyros as well as attitude sensors. These are the systems to which we feel that Kalman filtering is most ap- plicable. The Kalman filter uses a dynamical model for the time development of the system and a model of the sensor measurements to obtain the most accurate estimate possible of the system state using a linear estimator based on present and past measurements. It is, thus, ideally suited to both ground-based and on-board attitude determination. However, the applicability of the Kalman filtering technique rests on the availability of an accurate dynamical model. The dynamic equations for the spacecraft attitude pose many difficulties in the filter modeling. In particular, the external torques and the distribution of momentum internally due to the use of rotating or rastering instruments lead to significant uncertainties in the modeling. For autonomous spacecraft the use of inertial reference units as a model replacement permits the circumvention of these problems. In this representation the angular velocity of the spacecraft is obtained from the gyro data. The kinematic equations are used to obtain the attitude state and this is augmented by means of additional state-vector components for the gyro biases. Thus, gyro data are not treated as observations and the gyro noise appears as state noise rather than as observation noise. It is theoretically possible that a spacecraft is three-axis stabilized with such rigidity that the time development of the system can be described accurately without gyro information, or that it is one-axis stabilized so that only a single gyro is needed to provide information on the time history of the system. The modification of the algorithms presented here in order to apply to those cases is slight. However, this is of little practical importance because a control system capable of such

Optimal Estimation of Dynamic Systems

Abstract: LEAST SQUARES APPROXIMATION A Curve Fitting Example Linear Batch Estimation Linear Least Squares Weighted Least Squares Constrained Least Squares Linear Sequential Estimation Nonlinear Least Squares Estimation Basis Functions Advanced Topics Matrix Decompositions in Least Squares Kronecker Factorization and Least Squares Levenberg-Marquardt Method Projections in Least Squares Summary PROBABILITY CONCEPTS IN LEAST SQUARES Minimum Variance Estimation Estimation without a Prior State Estimates Estimation with a Prior State Estimates Unbiased Estimates Maximum Likelihood Estimation Cramer-Rao Inequality Nonuniqueness of the Weight Matrix Bayesian Estimation Advanced Topics Analysis of Covariance Errors Ridge Estimation Total Least Squares Summary REVIEW OF DYNAMICAL SYSTEMS Linear System Theory The State Space Approach Homogeneous Linear Dynamical Systems Forced Linear Dynamical Systems Linear State Variable Transformations Nonlinear Dynamical Systems Parametric Differentiation Observability Discrete-Time Systems Stability of Linear and Nonlinear Systems Attitude Kinematics and Rigid Body Dynamics Attitude Kinematics Rigid Body Dynamics Spacecraft Dynamics and Orbital Mechanics Spacecraft Dynamics Orbital Mechanics Aircraft Flight Dynamics Vibration Summary PARAMETER ESTIMATION: APPLICATIONS Global Positioning System Navigation Attitude Determination Vector Measurement Models Maximum Likelihood Estimation Optimal Quaternion Solution Information Matrix Analysis Orbit Determination Aircraft Parameter Identification Eigen-system Realization Algorithm Summary SEQUENTIAL STATE ESTIMATION A Simple First-Order Filter Example Full-Order Estimators Discrete-Time Estimators The Discrete-Time Kalman Filter Kalman Filter Derivation Stability and Joseph's Form Information Filter and Sequential Processing Steady-State Kalman Filter Correlated Measurement and Process Noise Orthogonality Principle The Continuous-Time Kalman Filter Kalman Filter Derivation in Continuous Time Kalman Filter Derivation from Discrete Time Stability Steady-State Kalman Filter Correlated Measurement and Process Noise The Continuous-Discrete Kalman Filter Extended Kalman Filter Advanced Topics Factorization Methods Colored-Noise Kalman Filtering Consistency of the Kalman Filter Adaptive Filtering Error Analysis Unscented Filtering Robust Filtering Summary BATCH STATE ESTIMATION Fixed-Interval Smoothing Discrete-Time Formulation Continuous-Time Formulation Nonlinear Smoothing Fixed-Point Smoothing Discrete-Time Formulation Continuous-Time Formulation Fixed-Lag Smoothing Discrete-Time Formulation Continuous-Time Formulation Advanced Topics Estimation/Control Duality Innovations Process Summary ESTIMATION OF DYNAMIC SYSTEMS: APPLICATIONS GPS Position Estimation GPS Coordinate Transformations Extended Kalman Filter Application to GPS Attitude Estimation Multiplicative Quaternion Formulation Discrete-Time Attitude Estimation Murrell's Version Farrenkopf's Steady-State Analysis Orbit Estimation Target Tracking of Aircraft The a-b Filter The a-b-g Filter Aircraft Parameter Estimation Smoothing with the Eigen-system Realization Algorithm Summary OPTIMAL CONTROL AND ESTIMATION THEORY Calculus of Variations Optimization with Differential Equation Constraints Pontryagin's Optimal Control Necessary Conditions Discrete-Time Control Linear Regulator Problems Continuous-Time Formulation Discrete-Time Formulation Linear Quadratic-Gaussian Controllers Continuous-Time Formulation Discrete-Time Formulation Loop Transfer Recovery Spacecraft Control Design Summary APPENDIX A MATRIX PROPERTIES Basic Definitions of Matrices Vectors Matrix Norms and Definiteness Matrix Decompositions Matrix Calculus APPENDIX B BASIC PROBABILITY CONCEPTS Functions of a Single Discrete-Valued Random Variable Functions of Discrete-Valued Random Variables Functions of Continuous Random Variables Gaussian Random Variables Chi-Square Random Variables Propagation of Functions through Various Models Linear Matrix Models Nonlinear Models APPENDIX C PARAMETER OPTIMIZATION METHODS C.1 Unconstrained Extrema C.2 Equality Constrained Extrema C.3 Nonlinear Unconstrained Optimization C.3.1 Some Geometrical Insights C.3.2 Methods of Gradients C.3.3 Second-Order (Gauss-Newton) Algorithm APPENDIX D COMPUTER SOFTWARE Index

The square-root unscented Kalman filter for state and parameter-estimation

Abstract: Over the last 20-30 years, the extended Kalman filter (EKF) has become the algorithm of choice in numerous nonlinear estimation and machine learning applications. These include estimating the state of a nonlinear dynamic system as well estimating parameters for nonlinear system identification (eg, learning the weights of a neural network). The EKF applies the standard linear Kalman filter methodology to a linearization of the true nonlinear system. This approach is sub-optimal, and can easily lead to divergence. Julier et al. (1997), proposed the unscented Kalman filter (UKF) as a derivative-free alternative to the extended Kalman filter in the framework of state estimation. This was extended to parameter estimation by Wan and Van der Merwe et al., (2000). The UKF consistently outperforms the EKF in terms of prediction and estimation error, at an equal computational complexity of (OL/sup 3/)/sup l/ for general state-space problems. When the EKF is applied to parameter estimation, the special form of the state-space equations allows for an O(L/sup 2/) implementation. This paper introduces the square-root unscented Kalman filter (SR-UKF) which is also O(L/sup 3/) for general state estimation and O(L/sup 2/) for parameter estimation (note the original formulation of the UKF for parameter-estimation was O(L/sup 3/)). In addition, the square-root forms have the added benefit of numerical stability and guaranteed positive semi-definiteness of the state covariances.