The history information space

The history I-space, $\mathcal I_{hist}$ , is essentially the set of all data the robots may ever obtain. Formally, for a time period $[t_0, t_f] \subset T$ , a perfect description of everything that occurred would be a state trajectory $\tilde{x}_t: [t_0, t_f] \to X$ , in which $X$ is the combined state space of robots and targets. It is impossible to obtain this because not all target positions are known. What is available is the robots' trajectory $\tilde{q}_t = \tau$ and the sensor observation history $\tilde{y}_t: [t_0, t_f] \to Y$ , produced by a sensor mapping $h: X \to Y$ , in which $Y$ is the observation space of the sensors. That is, $\tilde{y}_t$ is a time-parameterized collection of sensor observations. Let the robots also have access to some initial information $\eta_0$ at $t = t_0$ . The history I-state at time $t$ , $\eta_t = (\eta_0, \tilde{q}_t, \tilde{y}_t)$ , represents all information available to the robots. The history I-space $\mathcal I_{hist}$ is the set of all possible history I-states.

$\mathcal I_{hist}$ is an unwieldy space that must be greatly reduced if we expect to solve interesting problems. Imagine a robot equipped with a GPS and a video camera moves along some path $\tau$ . Without a specific task, the robot will not be able to decide what information it gathers is useful; therefore, it has to store all of $\tilde{q}_t, \tilde{y}_t$ . Even at a relatively low spatial resolution and a frequency of 30 Hz, just keeping the robot's locations and the camera's images in compressed form requires a large amount of storage space, which presently is not generally possible over a long time period.