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.