Distance between two sets

For the Boolean-valued function $ \phi $, there is no information about how far the robot is from hitting the obstacles. Such information is very important in planning algorithms. A distance function provides this information and is defined as $ d : {\cal C}
\rightarrow [0,\infty)$, in which the real value in the range of $ f$ indicates the distance in the world, $ {\cal W}$, between the closest pair of points over all pairs from $ {\cal A}(q)$ and $ {\cal O}$. In general, for two closed, bounded subsets, $ E$ and $ F$, of $ {\mathbb{R}}^n$, the distance is defined as

$\displaystyle \rho(E,F) = \min_{e \in E} \Big\{ \min_{f \in F} \Big\{ \Vert e - f\Vert \Big\} \Big\} ,$ (5.28)

in which $ \Vert\cdot\Vert$ is the Euclidean norm. Clearly, if $ E \cap F
\not = \emptyset$, then $ \rho(E,F) = 0$. The methods described in this section may be used to either compute distance or only determine whether $ q \in {\cal C}_{obs}$. In the latter case, the computation is often much faster because less information is required.

Steven M LaValle 2012-04-20