Chapter 8: KALMAN FILTERING

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 accelerometers and gyroscopes) or auxiliary sensors such as speed sensors (e.g., wheel speed sensors for land vehicles, water speed sensors for ships, air speed sensors for aircraft, or Doppler radar), magnetic compasses, altimeters (barometric or radar), or radionavigation aids (e.g., DME, VOR, LORAN).
The system state in question may include the position, velocity, acceleration, attitude, and attitude rate of a vehicle on land, at sea, in the air, or in space. The system state may also include ancillary "nuisance variables" for modeling time-correlated noise sources such as ionospheric propagation delays of GPS signals, and time-varying parameters of the sensors, GPS receiver clock frequency and phase, or scale factors and output biases of accelerometers or gyroscopes.
Uncertain dynamics includes unpredictable disturbances of the host vehicle, whether caused by a human operator or by the medium (e.g., winds, surface currents, turns in the road, or terrain changes), but it may also include unpredictable changes in the sensor parameters.

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