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Basic Notation

To make this more formal, let us denote the position (or: location) of a mobile robot by a three-dimensional variable tex2html_wrap_inline2837 , comprising its x-y coordinates (in some Cartesian coordinate system) and its heading direction tex2html_wrap_inline2843 . Let tex2html_wrap_inline2845 denote the robot's true location at time t, and tex2html_wrap_inline2849 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 tex2html_wrap_inline2851 denote the robot's position belief at time t. tex2html_wrap_inline2851 is a probability distribution over the space of positions. For example, tex2html_wrap_inline2857 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 tex2html_wrap_inline2875 (with tex2html_wrap_inline2877 ) 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.

Dieter Fox
Fri Nov 19 14:29:33 MET 1999