The ideal way to define distance on is to use a cost functional
and then define the distance from
to
as
the optimal cost-to-go from to while remaining in
. In some cases, it has been also referred to as the
*nonholonomic metric*, *Carnot-Caratheodory metric*, or
*sub-Riemannian metric* [596]. Note that this not a
true metric, as mentioned in Section 5.1.2, because the
cost may not be symmetric. For example, traveling a small distance
forward with Dubins car is much shorter than traveling a small
distance backward. If there are obstacles, it may not even be
possible to reach configurations behind the car.

This concept of distance should be somewhat disturbing because it
requires optimally solving the motion planning problem of Formulation
14.1. Thus, it cannot be practical for efficient use in a
motion planning algorithm. Nevertheless, understanding this ideal
notion of distance can be very helpful in designing practical distance
functions on . For example, rather than using a weighted Euclidean
metric (often called *Mahalanobis metric*) for the Dubins car, a
distance function can be defined based on the length of the shortest
path between two configurations. These lengths are straightforward to
compute, and are based on the optimal curve families that will be
covered in Section 15.3. This distance function
neglects obstacles, but it should still provide better distance
information than the weighted Euclidean metric. It may also be useful
for car models that involve dynamics.

The general idea is to get as close as possible to the optimal cost-to-go without having to perform expensive computations. It is often possible to compute a useful underestimate of the optimal cost-to-go by neglecting some of the constraints, such as obstacles or dynamics. This may help in applying search heuristics.

Steven M LaValle 2020-08-14