Overview

Heretofore, we have assumed that the planner has access either to an exact geometric description of its environment, or to a suite of sensors (e.g., sonars) that provide perfect information about the environment. In this chapter, we begin to consider cases for which the robot's knowledge of the world derives from measurements provided by imperfect, noisy sensors.

The Kalman filter is one of the most useful estimation tools available today. Loosely speaking, Kalman filtering provides a recursive method of estimating the state of a dynamical system in the presence of noise [215, 313]. A key feature of the Kalman filter is that it simultaneously maintains estimates of both the state vector and the estimate error covariance matrix ( P), which is equivalent to saying that the output of a Kalman filter is a Gaussian probability density function (PDF) with mean and covariance P. In the context of localization, the Kalman filter output is then a distribution of likely robot positions instead of a single position estimate. As such, the Kalman filter is a specific example of a more general technique known as probabilistic estimation. Some of more general probabilistic estimation techniques are presented in chapter 9.

We begin this chapter by presenting a conceptual overview of probabilistic estimation in section 8.1. Section 8.2 carefully derives the Kalman filter for linear systems. The approach taken here begins with a simplified version of the problem, then gradually adds complexity until the full Kalman filtering equations are reached.