Global Positioning Systems, Inertial Navigation, and Integration, Second Edition

8

KALMAN FILTERING

8.1 INTRODUCTION

Kalman’s paper introducing his now-famous filter was first published in 1960 [104], and its first practical implementation was for integrating an inertial navigator with airborne radar aboard the C5A military aircraft [137].

The application of interest here is quite similar. We want to integrate an onboard inertial navigator with a different electromagnetic ranging system (GPS). There are many ways to do this [18], but nearly all involve Kalman filtering.

The purpose of this chapter is to familiarize you with theoretical and practical aspects of Kalman filtering that are important for GPS/INS integration, and the presentation is primarily slanted toward this application. We have also included a brief derivation of the Kalman gain matrix, based on the maximum-likelihood estimation (MLE) model. Broader treatments of the Kalman filter are presented in Refs. 6, 30, 59, and 101; more basic introductions can be found in Refs. 48 and 218, more mathematically rigorous derivations can be found in Ref. 99; and more extensive coverage of the practical aspects of Kalman filtering can be found in Refs. 29 and 66.

8.1.1 What Is a Kalman Filter?

The Kalman filter is an extremely effective and versatile procedure for combining noisy sensor outputs to estimate the state of a system with uncertain dynamics, where

The noisy sensors could be just GPS receivers and inertial navigation systems, but may also include subsystem-level sensors (e.g., GPS clocks or INS ...