The continuous-space methods from Section 10.6 can be
directly applied to produce an approximate solution by interpolating
over to determine cost-to-go values. The initial cost-to-go
value over the collection of samples is obtained by
(12.6). Following (10.46), the dynamic
programming recurrence is
|
(12.10) |
If
is finite, the probability mass is
distributed over a finite set of points,
. This in turn implies that
is also distributed over a finite
subset of . This is somewhat unusual because is a
continuous space, which ordinarily requires the specification of a
probability density function. Since the set of future states is
finite, this enables a sum to be used in (12.10) as opposed
to an integral over a probability density function. This technically
yields a probability density over , but this density must
be expressed using Dirac functions.12.1 An approximation is still needed, however, because the
points may not be exactly the sample points on which the
cost-to-go function is represented.
Steven M LaValle
2020-08-14