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

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