11.2.1 Information Mappings

Figure 11.3: Many alternative information mappings may be proposed. Each leads to a derived information space.
\begin{figure}\centerline{\psfig{figure=figs/imaps.eps,width=4.0truein} }\end{figure}

Consider a function that maps the history I-space into a space that is simpler to manage. Formally, let $ {\kappa}: {\cal I}_{hist}\rightarrow
{\cal I}_{der}$ denote a function from a history I-space, $ {\cal I}_{hist}$, to a derived I-space, $ {\cal I}_{der}$. The function, $ {\kappa }$, is called an information mapping, or I-map. The derived I-space may be any set; hence, there is great flexibility in defining an I-map.11.2 Figure 11.3 illustrates the idea. The starting place is $ {\cal I}_{hist}$, and mappings are made to various derived I-spaces. Some generic mappings, $ {\kappa}_1$, $ {\kappa}_2$, and $ {\kappa}_3$, are shown, along with some very important kinds, $ {\cal I}_{est}$, $ {\cal I}_{ndet}$ and $ {\cal I}_{prob}$. The last two are the subjects of Sections 11.2.2 and 11.2.3, respectively. The other important I-map is $ {\kappa}_{est}$, which uses the history to estimate the state; hence, the derived I-space is $ X$ (see Example 11.11). In general, an I-map can even map any derived I-space to another, yielding $ {\kappa}: {\cal I}_{der}\rightarrow {\cal I}'_{der}$, for any I-spaces $ {\cal I}_{der}$ and $ {\cal I}'_{der}$. Note that any composition of I-maps yields an I-map. The derived I-spaces $ {\cal I}_2$ and $ {\cal I}_3$ from Figure 11.3 are obtained via compositions.

Steven M LaValle 2012-04-20