Due to the fundamental importance of numerical integration and the intricate link between discrepancy and integration error, most sampling literature has led to low-discrepancy sequences and point sets [738,893,937]. Although motion planning is quite different from integration, it is worth evaluating these carefully constructed and well-analyzed samples. Their potential use in motion planning is no less reasonable than using pseudorandom sequences, which were also designed with a different intention in mind (satisfying statistical tests of randomness).

Low-discrepancy sampling methods can be divided into three categories:
1) Halton/Hammersley sampling; 2) (t,s)-sequences and (t,m,s)-nets;
and 3) lattices. The first category represents one of the earliest
methods, and is based on extending the van der Corput sequence.
The *Halton sequence* is an -dimensional generalization of the
van der Corput sequence, but instead of using binary representations,
a different basis is used for each coordinate [430]. The
result is a reasonable deterministic replacement for random samples in
many applications. The resulting discrepancy (and dispersion) is
lower than that for random samples (with high probability). Figure
5.8a shows the first Halton points in
.

Choose relatively prime integers (usually the first primes, , , , are chosen). To construct the th sample, consider the base- representation for , which takes the form . The following point in is obtained by reversing the order of the bits and moving the decimal point (as was done in Figure 5.2):

(5.24) |

For , this yields the th point in the van der Corput sequence. Starting from , the th sample in the Halton sequence is

(5.25) |

Suppose instead that , the required number of points, is known. In
this case, a better distribution of samples can be obtained. The
*Hammersley point set* [431] is an adaptation of the Halton
sequence. Using only distinct primes and starting at ,
the th sample in a Hammersley point set with elements is

(5.26) |

Figure 5.8b shows the Hammersley set for and .

The construction of Halton/Hammersley samples is simple and efficient, which has led to widespread application. They both achieve asymptotically optimal discrepancy; however, the constant in their asymptotic analysis increases more than exponentially with dimension [738]. The constant for the dispersion also increases exponentially, which is much worse than for the methods of Section 5.2.3.

Improved constants are obtained for sequences and finite points by
using (t,s)-sequences, and (t,m,s)-nets, respectively
[738]. The key idea is to enforce zero discrepancy over
particular subsets of
known as *canonical rectangles*, and
all remaining ranges in
will contribute small amounts to
discrepancy. The most famous and widely used (t,s)-sequences are
Sobol' and Faure (see
[738]). The Niederreiter-Xing
(t,s)-sequence has the best-known
asymptotic constant, , in which is a small positive
constant [739].

The third category is *lattices*, which can be
considered as a generalization of grids that allows nonorthogonal axes
[682,893,959]. As an example, consider Figure
5.5b, which shows lattice points generated by the
following technique. Let be a positive irrational number.
For a fixed , generate the th point according to
, in which denotes the fractional part of the
real value (modulo-one arithmetic). In Figure 5.5b,
, the *golden
ratio*. This procedure can be generalized to
dimensions by picking distinct irrational numbers. A technique
for choosing the parameters by using the roots of irreducible
polynomials is discussed in [682]. The th sample in the
lattice is

(5.27) |

Recent analysis shows that some lattice sets achieve asymptotic discrepancy that is very close to that of the best-known nonlattice sample sets [445,938]. Thus, restricting the points to lie on a lattice seems to entail little or no loss in performance, but has the added benefit of a regular neighborhood structure that is useful for path planning. Historically, lattices have required the specification of in advance; however, there has been increasing interest in extensible lattices, which are infinite sequences [446,938].

Steven M LaValle 2020-08-14