Nonlinear state estimation using an invariant unscented Kalman filter
Jean-Philippe Condomines, Cédric Seren, Gautier Hattenberger · AIAA Guidance, Navigation, and Control (GNC) Conference · 2013
In this paper, we proposed a novel approach for nonlinear state estimation, named pi-IUKF (Invariant Unscented Kalman Filter), which is based on both invariant filter es-timation and UKF theoretical principles. Several research works on nonlinear invariant observers have been led and provide a geometrical-based constructive method for design-ing filters dedicated to nonlinear state estimation problems while preserving the physical properties and systems symmetries. The general invariant observer guarantees a straight-forward form of the nonlinear estimation error dynamics whose properties are remarkable. The developed pi-IUKF estimator suggests a systematic approach to determine all the symmetry-preserving correction terms, associated with a nonlinear state-space representa-tion 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 pi-IUKF converge to constant values due to the symmetry-preserving property provided by the nonlinear invariant estimation theory. The designed pi-IUKF method has been successfully applied to some relevant practical prob-lems 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). Nomenclature ei Orthonormal basis vector, i ∈ [ [ 1; 3]] ω Instantaneous angular velocity vector, rad/s ω0 Reference angular velocity vector in R3, rad/s a0, b0 Positive scalars q0 Unit quaternion φ Bank angle θ Pitch angle ψ Heading angle u Longitudinal speed component (body frame) v Lateral speed component (body frame) w Vertical speed component (body frame) A Constant gravity vector in the North-East-Down (NED) coordinate system i.e., A = ge3,m/s 2 a Specific acceleration vector, m/s2