5 research outputs found
Path Planning with Divergence-Based Distance Functions
Distance functions between points in a domain are sometimes used to
automatically plan a gradient-descent path towards a given target point in the
domain, avoiding obstacles that may be present. A key requirement from such
distance functions is the absence of spurious local minima, which may foil such
an approach, and this has led to the common use of harmonic potential
functions. Based on the planar Laplace operator, the potential function
guarantees the absence of spurious minima, but is well known to be slow to
numerically compute and prone to numerical precision issues. To alleviate the
first of these problems, we propose a family of novel divergence distances.
These are based on f-divergence of the Poisson kernel of the domain. We define
the divergence distances and compare them to the harmonic potential function
and other related distance functions.
Our first result is theoretical: We show that the family of divergence
distances are equivalent to the harmonic potential function on simply-connected
domains, namely generate paths which are identical to those generated by the
potential function. The proof is based on the concept of conformal invariance.
Our other results are more practical and relate to two special cases of
divergence distances, one based on the Kullback-Leibler divergence and one
based on the total variation divergence. We show that using divergence
distances instead of the potential function and other distances has a
significant computational advantage, as, following a pre-processing stage, they
may be computed up to an order of magnitude faster than the others when taking
advantage of certain sparsity properties of the Poisson kernel. Furthermore,
the computation is "embarrassingly parallel", so may be implemented on a GPU
with up to three orders of magnitude speedup
Practical Distance Functions for Path-Planning in Planar Domains
Path planning is an important problem in robotics. One way to plan a path
between two points within a (not necessarily simply-connected) planar
domain , is to define a non-negative distance function on
such that following the (descending) gradient of this
distance function traces such a path. This presents two equally important
challenges: A mathematical challenge -- to define such that has a
single minimum for any fixed (and this is when ), since a local
minimum is in effect a "dead end", A computational challenge -- to define
such that it may be computed efficiently. In this paper, given a description of
, we show how to assign coordinates to each point of and
define a family of distance functions between points using these coordinates,
such that both the mathematical and the computational challenges are met. This
is done using the concepts of \emph{harmonic measure} and
\emph{-divergences}.
In practice, path planning is done on a discrete network defined on a finite
set of \emph{sites} sampled from , so any method that works well on the
continuous domain must be adapted so that it still works well on the discrete
domain. Given a set of sites sampled from , we show how to define a
network connecting these sites such that a \emph{greedy routing} algorithm
(which is the discrete equivalent of continuous gradient descent) based on the
distance function mentioned above is guaranteed to generate a path in the
network between any two such sites. In many cases, this network is close to a
(desirable) planar graph, especially if the set of sites is dense