11.4.1 Discrete-Stage Information Spaces

Assume here that there are discrete stages. Let
be an -dimensional manifold for called the *state space*.^{11.4} Let
be an
-dimensional manifold for
called the *observation space*. For each , let
be an -dimensional manifold for
called the set of
*nature sensing actions*. The three
kinds of sensors mappings, , defined in Section 11.1.1
are possible, to yield either a *state mapping*, , a
*state-nature mapping*
, or a *history-based*,
. For the case of a state mapping, the
preimages, , once again induce a partition of . Preimages
can also be defined for state-action mappings, but they do not
necessarily induce a partition of .

Many interesting sensing models can be formulated in continuous state
spaces. Section 11.5.1 provides a kind of sensor catalog.
There is once again the choice of nondeterministic or probabilistic
uncertainty if nature sensing actions are used. If nondeterministic
uncertainty is used, the expressions are the same as the discrete
case. Probabilistic models are defined in terms of a probability
density function,
,^{11.5} in which denotes the continuous-time
replacement for . The model can also be expressed as
, in that same manner that was obtained for discrete
state spaces.

The usual three choices exist for the initial conditions: 1) Either
is given; 2) a subset is given; or 3) a
probability density function, , is given. The initial
condition spaces in the last two cases can be enormous. For example,
if and any subset is possible as an initial condition,
then
, which has higher cardinality than
. If
any probability density function is possible, then
is a space
of functions.^{11.6}

The I-space definitions from Section 11.1.2 remain the same, with the understanding that all of the variables are continuous. Thus, (11.17) and (11.19) serve as the definitions of and . Let be an -dimensional manifold for . For each and , let be an -dimensional manifold for . A discrete-stage I-space planning problem over continuous state spaces can be easily formulated by replacing each discrete variable in Formulation 11.1 by its continuous counterpart that uses the same notation. Therefore, the full formulation is not given.

Steven M LaValle 2020-08-14