Abstract-In this paper, we propose a systematization of the (discrete-time) Unscented Kalman Filter (UKF) theory. We gather all available UKF variants in the literature, present corrections to theoretical inconsistencies, and provide a tool for the construction of new UKF's in a consistent way. This systematization is done, mainly, by revisiting the concepts of Sigma-Representation, Unscented Transformation (UT), Scaled Unscented Transformation (SUT), UKF, and Square-Root Unscented Kalman Filter (SRUKF). Inconsistencies are related to 1) matching the order of the transformed covariance and cross-covariance matrices of both the UT and the SUT; 2) multiple UKF definitions; 3) issue with some reduced sets of sigma points described in the literature; 4) the conservativeness of the SUT; 5) the scaling effect of the SUT on both its transformed covariance and crosscovariance matrices; and 6) possibly ill-conditioned results in SRUKF's. With the proposed systematization, the symmetric sets of sigma points in the literature are formally justified, and we are able to provide new consistent variations for UKF's, such as the Scaled SRUKF's and the UKF's composed by the minimum number of sigma points. Furthermore, our proposed SRUKF has improved computational properties when compared to state-ofthe-art methods.

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Abstract-The paper presents an application of unscented Kalman filters to an automotive electronic throttle device. The motivation of this study is on estimating the position of the throttle device when measurements of position are inaccessible, e.g., due to failures in the sensor of position. In this case, an external wattmeter is connected in the circuitry to measure the power consumed by the throttle, and this information feeds unscented Kalman filters to produce the estimation for the position. Experimental data support the findings of this paper.Note to Practitioners: Abstract-Almost all of the brand-new vehicles based on spark-ignition combustion engines have an electronic throttle valve to control the power produced by the engine. The electronic throttle has a unique sensor for measuring the position of the throttle valve, and this feature can represent a serious problem when the sensor of position fails. As an attempt to prevent the effects of a failure from such sensor, we present an algorithm (unscented Kalman filter) combined with the use of an additional sensor, a wattmeter. The wattmeter is detached from the throttle's structure but is arranged so as to measure the electric power consumed by the throttle. Measurements of the power consumption then feed the unscented Kalman filter-this filter then produces an estimation of the position of the throttle valve. Experimental data illustrate the practical benefits of our approach.

Unscented Kalman Filters (UKFs) have become popular in the research community. Most UKFs work only with Euclidean systems, but in many scenarios it is advantageous to consider systems with state-variables taking values on Riemannian manifolds. However, we can still find some gaps in the literature's theory of UKFs for Riemannian systems: for instance, the literature has not yet i) developed Riemannian extensions of some fundamental concepts of the UKF theory (e.g., extensions of σ-representation, Unscented Transformation, Additive UKF, Augmented UKF, additive-noise system), ii) proofs of some steps in their UKFs for Riemannian systems (e.g., proof of sigma points parameterization by vectors, state correction equations, noise statistics inclusion), and iii) relations between their UKFs for Riemannian systems. In this work, we attempt to develop a theory capable of filling these gaps. Among other results, we propose Riemannian extensions of the main concepts in the UKF theory (including closed forms), justify all steps of the proposed UKFs, and provide a framework able to relate UKFs for particular manifolds among themselves and with UKFs for Euclidean spaces. Compared with UKFs for Riemannian manifolds of the literature, the proposed filters are more consistent, formallyprincipled, and general. An example of satellite attitude tracking illustrates the proposed theory. Riemannian systems: the works [34], [35] and [36] introduced EKFs for Lie groups; and [5] a UKF for geodesically-complete Riemannian manifolds.geodesically-complete

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Proper construction of an unscented Kalman filter (UKF) for unit quaternionic systems is not straightforward due to the incompatibility between the algebraic properties of the unit quaternions and the common real vector space operations (additions and scalar multiplications) needed in the steps of a filter algorithm. This work studies, in detail, all UKFs and square-root UKFs for quaternionic systems proposed in the literature. First, we classify the algorithms according to the preservation of the unity norm of the quaternion variables. Second, we propose two new algorithms: the quaternionic additive unscented Kalman filter (QuAdUKF) and a square-root variant of it. The QuAdUKF encompasses all known UKFs for quaternionic systems of the literature preserving, in all steps, the norm of the unit quaternion variables. Besides, it can also yield new UKFs with this norm preservation property. The QuAdUKF's square-root variant has better properties in comparison with all the square-root UKFs for quaternionic systems of the literature. Numerical experiments for a spacecraft attitude estimation problem illustrate the theoretical results. KEYWORDS attitude estimation, square-root unscented Kalman filter (SRUKF), unit quaternion, unscented Kalman filter (UKF)In the control literature, the most common state space models are those where the underlying state, input, and output variables are real vectors, that is, elements that lie in a Euclidean vector space ℝ n . For bodies under two-dimensional or three-dimensional (3D) motions, these Euclidean models fit for dimensionless material points, that is, linear displacements and velocities. However, for large rigid bodies, besides these linear displacement characteristics, the body pointing direction and angular (rotational) movements become important. [1][2][3] It is well known that rotations of rigid bodies in a 3D space are mathematically represented by state trajectories, restricted to a compact manifold called the special orthogonal group, SO(3). The high nonlinearity of this manifold, due to the restriction of variables in specific regions, leads to difficulties in dealing with these variables. In estimation algorithms, performing calculations with variables in SO(3) is often computationally expensive, and more computationally-efficient rotation parameterizations were proposed in the literature. Among the alternative rotation parameterizations, such as Euler angles, rotation vectors (RoVs), and unit quaternions, the unit quaternions are often chosen because, unlike other 4500

In this work, a new minimum set of sigma points for unscented filtering is proposed along with its unscented Kalman filter in both square-root and nonsquare-root forms. Comparative with the other reduced sigma sets of the literature, the new sigma set is, in some cases, better defined or, in another case, a generalization. In numerical examples, the unscented transform of the new sigma set is compared with the unscented transforms of the other reduced sigma sets of the literature. In addition, the performance of the new unscented Kalman filter is studied in an aircraft target tracking scenario.

This paper introduces a filter for tracking a ballistic missile during its boost-phase. This filter includes a new switching algorithm and a modified Interacting Multiple Model Unscented Filter (IMMUF) where the Markov Transition Matrix is time-variable. Position, velocity and all unknown parameters of a medium-range ballistic missile model are reconstructed. Simulations demonstrate the new filter is able to consistently estimate a missile's trajectory and all unknown parameters and to outperform previous forms of the IMMUF.