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5.2.1 Motivation and Basic Concepts

The state space for motion planning, , is uncountably infinite,
yet a sampling-based planning algorithm can consider at most a
countable number of samples. If the algorithm runs forever, this may
be countably infinite, but in practice we expect it to terminate
early after only considering a finite number of samples. This
mismatch between the cardinality of and the set that can be
probed by an algorithm motivates careful consideration of sampling
techniques. Once the sampling component has been defined, discrete
planning methods from Chapter 2 may be adapted to the
current setting. Their performance, however, hinges on the way the
C-space is sampled.

Since sampling-based planning algorithms are often terminated early,
the particular order in which samples are chosen becomes critical.
Therefore, a distinction is made between a sample *set* and a sample *sequence*. A unique sample set can always be constructed from a
sample sequence, but many alternative sequences can be constructed
from one sample set.

**Subsections**
Steven M LaValle
2012-04-20