A simple way to handle obstacles is to determine for each
whether
. This can be computed and stored in an array
before the value iterations are performed. For rigid robots, this can
be efficiently computed using *fast Fourier transforms*
[513]. For each
,
. No value
iterations are performed on these states; their values must remain at
infinity. During the evaluation of (8.59) (or a
higher dimensional version), different actions are attempted. For
each action, it is required that all of the interpolation neighbors of
lie in
. If one of them lies in
, then that
action produces infinite cost. This has the effect of automatically
reducing the interpolation region, , to all cubes whose vertices
all lie in
, as shown in Figure 8.21b. All samples
in
are assumed to be deleted from in the remainder of this
section; however, the full grid is still used for interpolation so
that infinite values represent the obstacle region.

Note that as expressed so far, it is possible that points in may lie in because collision detection is performed only on the samples. In practice, either the grid resolution must be made fine enough to minimize the chance of this error occurring or distance information from a collision detection algorithm must be used to infer that a sufficiently large ball around each sample is collision free. If an interpolation region cannot be assured to lie in , then the resolution may have to be increased, at least locally.

Steven M LaValle 2020-08-14