Major Development Under Gaussian Filtering Since Unscented Kalman Filter Abhinoy Kumar Singh, Member, IEEE Abstract—Filtering is a recursive estimation of hidden states of a dynamic system from noisy measurements. Such problems appear in several branches of science and technology, ranging from target tracking to biomedical monitoring. A commonly practiced approach of filtering with nonlinear systems is Gaussian filtering. The early Gaussian filters used a derivative-based implementation, and suffered from several drawbacks, such as the smoothness requirements of system models and poor stability. A derivative-free numerical approximation-based Gaussian filter, named the unscented Kalman filter (UKF), was introduced in the nineties, which offered several advantages over the derivative- based Gaussian filters. Since the proposition of UKF, derivative- free Gaussian filtering has been a highly active research area. This paper reviews significant developments made under Gaussian filtering since the proposition of UKF. The review is particularly focused on three categories of developments: i) advancing the numerical approximation methods; ii) modifying the conventional Gaussian approach to further improve the filtering performance; and iii) constrained filtering to address the problem of discrete-time formulation of process dynamics. This review highlights the computational aspect of recent developments in all three categories. The performance of various filters are analyzed by simulating them with real-life target tracking problems. Index Terms—Bayesian framework, cubature rule-based filtering, Gaussian filters, Gaussian sum and square-root filtering, nonlinear filtering, quadrature rule-based filtering, unscented transformation.   I. Introduction T HE modern era of science and technology has witnessed huge applications requiring state estimations from noisy measurements, e.g., target tracking, stochastic modeling, industrial diagnosis, and prognosis etc. [1]. Thanks to the filtering algorithms [1]–[5], which offer an efficient recursive state estimation tool. Filtering algorithms have helped in the modernization of several domains, such as space technology [1], medical [6], finance and economics [7], weather monitoring [8], etc. The Bayesian framework [9], [10] is a common choice among the practitioners dealing with real-life estimation and filtering problems. It gives a probabilistic estimate of states by formulating prior and posterior probability density functions (pdf) [9], [10]. An analytical simplification is perceived by analyzing these pdfs numerically, and the Gaussian filters [1], [10] are most popular in this practice because of their high estimation accuracy at low computational cost. The Gaussian filters approximate the pdfs (prior and posterior) as Gaussian, and characterize them with mean and covariance. Researchers and industrial practitioners engaged with estimation applications would be grateful to Rudolph E. Kalman, who developed an optimal state estimation and filtering technique [11] for linear systems with white Gaussian noises. In the filtering literature, this technique is popularly known as Kalman filter [1]–[4]. The Kalman filter can be implemented with several linearly approximated filtering problems in different domains, like in target tracking [12] and communication systems [13]. However, the linearity constraint restricts its application with many more practical problems, where the linear approximation gives a poor characterization of system dynamics. A nonlinear filter, named as extended Kalman filter (EKF) [1], [3], was developed in the latter half of the sixties. It introduced a derivative-based local linearization of system dynamics, and propagated the mean and covariance through the locally linearized system models. It was further modified to improve the filtering performance by introducing several variants [14]–[16]. However, the derivative-based local linearization introduces many disadvantages to the EKF and its variants, like the smoothness requirement for systems model, poor estimation accuracy, and low convergence rate [1]–[3]. Despite all these drawbacks, the EKF and its variants were the only alternative for more than three decades. × A derivative-free filtering method was introduced in the nineties, which is known as the unscented Kalman filter (UKF) [17]–[20]. The UKF approximates the desired pdfs as Gaussian, and characterizes them with mean and covariance. Although the Gaussian approximation of unknown pdfs encounters an error, it is more accurate compared to the EKF [17]–[20]. Furthermore, the mean and covariance are obtained from the first and second moments. The moment computation encounters an integral of the form “nonlinear function Gaussian distribution.” Such integrals are generally intractable, and approximated with numerical methods. These methods approximate the integrals up to a particular order of Taylor series expansion. Thus, the moment of higher-order terms is another source of approximation error or the estimation error. Full expression of mean in terms of higher- order moments is derived in Appendix A. Nevertheless, it Manuscript received October 12, 2019; revised February 4, 2020; accepted April 2, 2020. This work was supported by INSPIRE Faculty Award, Department of Science and Technology, Government of India. Recommended by Associate Editor Yebin Wang. Citation: A. K. Singh, “Major development under Gaussian filtering since unscented Kalman filter,” IEEE/CAA J. Autom. Sinica, vol. 7, no. 5, pp. 1308–1325, Sept. 2020. A. K. Singh is with the Department of Electrical Engineering, Indian Institute of Technology Indore, Simrol, Indore 453552, India (e-mail: abhinoy.singh@iiti.ac.in). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JAS.2020.1003303 1308 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 7, NO. 5, SEPTEMBER 2020