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.