Localization in a continuous space using probabilistic models has
received substantial attention in recent years
[258,447,622,825,887,962].
It is often difficult to localize mobile robots because of noisy
sensor data, modeling errors, and high demands for robust operation
over long time periods. Probabilistic modeling and the computation of
probabilistic I-states have been quite successful in many experimental
systems, both for indoor and outdoor mobile robots. Figure
12.11 shows localization successfully being solved using
sonars only. The vast majority of work in this context involves
passive localization because the robot is often completing some other
task, such as reaching a particular part of the environment.
Therefore, the focus is mainly on *computing* the probabilistic
I-states, rather than performing a difficult search on
.

Probabilistic localization in continuous spaces most often involves the definition of the probability densities and (in the case of a state sensor mapping). If the stages represent equally spaced times, then these densities usually remain fixed for every stage. The state space is usually to account for translation and rotation, but it may be for translation only. The density accounts for the unpredictability that arises when controlling a mobile robot over some fixed time interval. A method for estimating this distribution for nonholonomic robots by solving stochastic differential equations appears in [1004].

The density indicates the relative likelihood of various measurements when given the state. Most often this models distance measurements that are obtained from a laser range scanner, an array of sonars, or even infrared sensors. Suppose that a robot moves around in a 2D environment and takes depth measurements at various orientations. In the robot body frame, there are angles at which a depth measurement is taken. Ideally, the measurements should look like those in Figure 11.15b; however, in practice, the data contain substantial noise. The observation is an -dimensional vector of noisy depth measurements.

One common way to define is to assume that the error in each distance measurement follows a Gaussian density. The mean value of the measurement can easily be calculated as the true distance value once is given, and the variance should be determined from experimental evaluation of the sensor. If it is assumed that the vector of measurements is modeled as a set of independent, identically distributed random variables, a simple product of Guassian densities is obtained for .

Once the models have been formulated, the computation of probabilistic I-states directly follows from Sections 11.2.3 and 11.4.1. The initial condition is a probability density function, , over . The marginalization and Bayesian update rules are then applied to construct a sequence of density functions of the form for every stage, .

In some limited applications, the models used to express
and
may be linear and Gaussian. In
this case, the Kalman filter of Section 11.6.1 can be easily
applied. In most cases, however, the densities will not have this
form. Moment-based approximations, as discussed in Section
11.4.3, can be used to approximate the densities. If
second-order moments are used, then the so-called *extended Kalman
filter* is obtained, in which the Kalman filter update rules can be
applied to a *linear-Gaussian*
approximation to the original problem. In recent years, one of the
most widely accepted approaches in experimental mobile robotics is to
use sampling-based techniques to directly compute and estimate the
probabilistic I-states. The particle-filtering approach, described in
general in Section 11.6.2, appears to provide good
experimental performance when applied to localization. The
application of particle filtering in this context is often
referred to as *Monte Carlo localization*; see the references at
the end of this chapter.

Steven M LaValle 2012-04-20