Recursive Batch Gaussian Sum Filter for Single-Beacon Navigation Problem

Abstract

A problem of determining the position of a slave autonomous unmanned under-water vehicle (AUV) using range measurements to the leader AUV is considered within the framework of the Bayesian stochastic approach. This problem can be interpreted as a single-beacon navigation problem. Its key feature is that the form of the posterior probability density function (PDF) varies from multi-extremal to single-extremal one. Previously, this problem was solved with the use of the Recurrent Iterative Batch Linearized Smoothing Filter with Multiple Linearization (RI-BMLS) proposed by the authors. The essence of that approach lies in identifying the point in time at which the posterior PDF becomes single-extremal, and then the recursive iterative Kalman filter is used to solve the problem. However, the RI-BMLS has some significant drawbacks: first, it is ineffective until the onset of aforesaid point in time; and second, the RI-BMLS can be computationally intensive when the posterior PDF remains multi-extremal over a long-time interval. This paper proposes the Recursive Iterative Batch Gaussian Sum Filter (RI-BGSF), which is a modification of the RI-BMLS. Unlike the RI-BMLS, the proposed algorithm allows for the effective solution of the single-beacon navigation problem throughout the entire problem solution, which is made possible due to the representation of the posterior PDF at its multi-extremum points as a weighted set of Gaussian densities. The RI-BGSF is significantly much simpler computationally owing to the combined use of recursive and nonrecursive schemes in its design. The advantages of the proposed algorithm are demonstrated through simulations.