Some sampling-based algorithms require choosing motion directions at
random.^{5.4} From a configuration , the possible directions of
motion can be imagined as being distributed around a sphere. In an
-dimensional C-space, this corresponds to sampling on
.
For example, choosing a direction in
amounts to picking an
element of
; this can be parameterized as
. If , then the previously mentioned trick for
should be used. If or , then samples can be
generated using a slightly more expensive method that exploits
spherical symmetries of the multidimensional Gaussian density function
[341]. The method is explained for
; boundaries and
identifications must be taken into account for other spaces. For each
of the coordinates, generate a sample from a zero-mean
Gaussian distribution with the same variance for each coordinate.
Following from the Central Limit Theorem, can be
approximately obtained by generating samples at random over
and adding them ( is usually sufficient in
practice). The vector
gives a random
direction in
because each was obtained
independently, and the level sets of the resulting probability density
function are spheres. We did not use uniform random samples for each
because this would bias the directions toward the corners of a
cube; instead, the Gaussian yields spherical symmetry. The final step
is to normalize the vector by taking
for each
coordinate.

Steven M LaValle 2012-04-20