To make this more formal, let us denote the *position* (or: *
location*) of a mobile robot by a three-dimensional variable , comprising its *x*-*y* coordinates (in
some Cartesian coordinate system) and its heading direction .
Let denote the robot's true location at time *t*, and
denote the corresponding random variable. Throughout this paper, we
will use the terms position and location interchangeably.

Typically, the robot does not know its exact position. Instead, it
carries a belief as to where it might be. Let denote the
robot's position belief at time *t*. is a probability
distribution over the space of positions. For example,
is the probability (density) that the robot assigns to the possibility
that its location at time *t* is *l*. The belief is updated in
response to two different types of events: The arrival of a
measurement through the robot's environment sensors (e.g., a camera
image, a sonar scan), and the arrival of an odometry reading (e.g.,
wheel revolution count). Let us denote environment sensor
measurements by *s* and odometry measurements by *a*, and the
corresponding random variables by *S* and *A*, respectively.

The robot perceives a stream of measurements, sensor measurements *s*
and odometry readings *a*. Let

denote the stream of measurements, where each (with ) either is a sensor measurement or an odometry reading. The
variable *t* indexes the data, and *T* is the most recently collected
data item (one might think of *t* as ``time''). The set *d*, which
comprises all available sensor data, will be referred to as the *
data*.

Fri Nov 19 14:29:33 MET 1999