11.1.2 Defining the History Information Space

This section defines the most basic and natural I-space. Many others will be derived from it, which is the topic of Section 11.2. Suppose that $ X$, $ U$, and $ f$ have been defined as in Formulation 10.1, and the notion of stages has been defined as in Formulation 2.2. This yields a state sequence $ x_1$, $ x_2$, $ \ldots $, and an action sequence $ u_1$, $ u_2$, $ \ldots $, during the execution of a plan. However, in the current setting, the state sequence is not known. Instead, at every stage, an observation, $ y_k$, is obtained. The process depicted in Figure 11.2.

Figure 11.2: In each stage, $ k$, an observation, $ y_k \in Y$, is received and an action $ u_k \in U$ is applied. The state, $ x_k$, however, is hidden from the decision maker.
\xymatrix{y_1 & & y_2 & & y_3 & \\
x_1 \ar[u]...
...r[r] &
u_2 \ar[r] & x_3 \ar[u] \ar[r] & \ldots }

In previous formulations, the action space, $ U(x)$, was generally allowed to depend on $ x$. Since $ x$ is unknown in the current setting, it would seem strange to allow the actions to depend on $ x$. This would mean that inferences could be made regarding the state by simply noticing which actions are available.11.1Instead, it will be assumed by default that $ U$ is fixed for all $ x \in X$. In some special contexts, however, $ U(x)$ may be allowed to vary.

Steven M LaValle 2012-04-20