Discretization

Assume that $ {\cal C}$ is discretized by using the resolutions $ k_1$, $ k_2$,$ \ldots $, and $ k_n$, in which each $ k_i$ is a positive integer. This allows the resolution to be different for each C-space coordinate. Either a standard grid or a Sukharev grid can be used. Let

$\displaystyle \Delta q_i = [ 0 \;\; \cdots \;\; 0 \;\; 1/k_i \;\; 0 \;\; \cdots \;\; 0 ],$ (5.35)

in which the first $ i-1$ components and the last $ n-i$ components are 0. A grid point is a configuration $ q \in {\cal C}$ that can be expressed in the form5.10

$\displaystyle \sum_{i=1}^n j_i \Delta q_i ,$ (5.36)

in which each $ j_i \in \{ 0,1,\ldots,k_i\}$. The integers $ j_1$, $ \ldots $, $ j_n$ can be imagined as array indices for the grid. Let the term boundary grid point refer to a grid point for which $ j_i = 0$ or $ j_i = k_i$ for some $ i$. Due to identifications, boundary grid points might have more than one representation using (5.36).

Steven M LaValle 2020-08-14