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Almost immediately following its introduction in 1960, the Kalman filter and the extended Kalman filter have served as the primary algorithms used to solve navigation problems [1–3]. The optimal, recursive, and online characteristics of the algorithm are perfectly suited to serve a wide range of applications requiring real‐time navigation solutions.

The traditional Kalman filter and extended Kalman filter are based on the following assumptions:

 Linear (or nearly linear) system dynamics and observations.

 All noise and error sources are Gaussian.

While these assumptions are valid in many cases, there is increasing interest in incorporating sensors and systems that are non‐Gaussian, nonlinear, or both. Because these characteristics inherently violate the fundamental assumptions of the Kalman filter, when Kalman filters are used, performance suffers. More specifically, this can result in filter estimates that are inaccurate, inconsistent, or unstable. To address this limitation, researchers have developed a number of algorithms designed to provide improved performance for nonlinear and non‐Gaussian problems [4–6].

In this chapter, we provide an overview of some of the most common and useful classes of nonlinear recursive estimators. The goal is to introduce the fundamental theories supporting the algorithms, identify their associated performance characteristics, and finally present their respective applicability from a navigation perspective.

The chapter is organized as follows. First, an overview of the notation and essential concepts related to estimation and probability theory are presented as a foundation for nonlinear filtering development. Some of the concepts include recursive estimation frameworks, the implicit assumptions and limitations of traditional estimators, and the deleterious effects on performance when these assumptions are not satisfied. Next an overview of nonlinear estimation theory is presented with the goal of demonstrating and deriving three main classes of nonlinear recursive estimators. These include Gaussian sum filters, grid particle filters, and sampling particle filters. Each of these classes of nonlinear recursive estimators is demonstrated and evaluated using a simple navigation example. The chapter is concluded with a discussion regarding the strengths and weaknesses of the approaches discussed with an emphasis on helping navigation engineers decide which estimation algorithm to apply to a given problem of interest.