##

5.2.2 Random Sampling

Now imagine moving beyond and generating a dense sample
sequence for any bounded C-space,
. In this
section the goal is to generate *uniform random* samples. This
means that the probability density function over is
uniform. Wherever relevant, it also will mean that the probability
density is also consistent with the Haar measure. We will not allow
any artificial bias to be introduced by selecting a poor
parameterization. For example, picking uniform random Euler angles
does *not* lead to uniform random samples over . However,
picking uniform random unit quaternions works perfectly because
quaternions use the same parameterization as the Haar measure; both
choose points on
.

Random sampling is the easiest of all sampling methods to apply to
C-spaces. One of the main reasons is that C-spaces are formed from
Cartesian products, and independent random samples extend easily
across these products. If
, and uniform random
samples and are taken from and , respectively,
then is a uniform random sample for . This is very
convenient in implementations. For example, suppose the motion
planning problem involves robots that each translate for any
; this yields
. In this case,
points can be chosen uniformly at random from and
combined into a -dimensional vector. Samples generated this way
are uniformly randomly distributed over . Combining samples over
Cartesian products is much more difficult for nonrandom
(deterministic) methods, which are presented in Sections
5.2.3 and 5.2.4.

**Subsections**
Steven M LaValle
2020-08-14