Nonlinear state estimation using an invariant unscented Kalman filter

Abstract

In this paper, we proposed a novel approach for nonlinear state estimation, named π-IUKF (Invariant Unscented Kalman Filter), which is based on both invariant filter estimation and UKF theoretical principles. Several research works on nonlinear invariant observers have been led and provide a geometrical-based constructive method for designing filters dedicated to nonlinear state estimation problems while preserving the physical properties and systems symmetries. The general invariant observer guarantees a straightforward form of the nonlinear estimation error dynamics whose properties are remarkable. The developed π-IUKF estimator suggests a systematic approach to determine all the symmetry-preserving correction terms, associated with a nonlinear state-space representation used for prediction, without requiring any linearization of the differential equations. The exploitation of the UKF principles within the invariant framework has required the definition of a compatibility condition on the observation equations. As a first result, the estimated covariance matrices of the π-IUKF converge to constant values due to the symmetry-preserving property provided by the nonlinear invariant estimation theory. The designed π-IUKF method has been successfully applied to some relevant practical problems such as the estimation of Attitude and Heading for aerial vehicles using low-cost AH reference systems (i.e., inertial/magnetic sensors characterized by low performances).

Invariant unscented Kalman filter with application to attitude estimation

“…Nevertheless, the IEKF, and more generally invariant observers, are characterized by a larger convergence domain, due to the exploitation of systems' symmetries within the estimation algorithm (i.e., within filter equations and gains computation), and present very good performances in practice. In order to derive more tractable nonlinear invariant state estimation algorithms, motivated by the practical problems encountered by the authors with miniUAVs flight control and guidance, civil Aircraft modeling and identification and dynamic system fault detection, isolation and recovery, an hybridization of the Unscented KF (UKF) principles [20], [16], [19] with invariant observers theory has been recently proposed in [10], [11]. Among other things, it has been proved in these bibliographical references that an Invariant UKF-like estimator (named IUKF) could be simply designed by introducing both notions of invariant state estimation and invariant output errors within any UKF algorithm formulation, whatever this latter corresponds to the standard version of the algorithm or to some square-root/UD factorized ones.…”

“…Inspired by the theory of continuous-time symmetry preserving observer [7] a novel and original UKF-based approach has been developed in [12] to address the approximation issue of the invariant EKF without requiring any linearization of the dynamical systems equations or compatibility condition such as proposed in the π-IUKF algorithm [10], [11]. The IUKF relies on the basic theoretical principles developed by Julier and Uhlmann at the beginning of 2000s (see [16]) which have been since widely applied to various nonlinear state estimation problems (cf.…”

“…Let's consider the observer in [11] for the IUKF algorithm s.t. : The invariant state estimation error dynamics 9 ηpx t ,x t q is given by:…”

“…Although dynamical systems possessing symmetries have been studied in control theory, few results taking benefit of system invariances for observers design exist today. Invariant nonlinear estimation theory appears so as a young research area in which the first main contributions can be dated from the beginning of 2000s ( [1], [3], [2], [5], [4], [6], [8], [9], [15], [10], [11], [17], [18] This optimality must be considered here w.r.t. an invariant state estimation error which will be defined precisely further.…”

Pi-Invariant Unscented Kalman Filter for sensor fusion

“…Unfortunately, the resulting Invariant EKF (IEKF) is based on second-order approximations for the invariant estimation error (cf [14]). To overcome this issue, an approach, primary proposed in [15] and named π-IUKF, also dedicated to nonlinear systems possessing symmetries, has been developed and successfully applied to an attitude estimation problem. The results have highlighted that the π-IUKF had the same estimation invariance properties than the ones obtained with the IEKF.…”