Title: Quantizing Euclidean motions via double-coset decomposition,
Abstract: Concepts from mathematical crystallography and group theory are used here to
quantize the group of rigid-body motions, resulting in a "motion alphabet" with
which to express robot motion primitives. From these primitives it is possible
to develop a dictionary of physical actions. Equipped with an alphabet of the
sort developed here, intelligent actions of robots in the world can be
approximated with finite sequences of characters, thereby forming the
foundation of a language in which to articulate robot motion. In particular, we
use the discrete handedness-preserving symmetries of macromolecular crystals
(known in mathematical crystallography as Sohncke space groups) to form a
coarse discretization of the space $\rm{SE}(3)$ of rigid-body motions. This
discretization is made finer by subdividing using the concept of double-coset
decomposition. More specifically, a very efficient, equivolumetric quantization
of spatial motion can be defined using the group-theoretic concept of a
double-coset decomposition of the form $\Gamma \backslash \rm{SE}(3) / \Delta$,
where $\Gamma$ is a Sohncke space group and $\Delta$ is a finite group of
rotational symmetries such as those of the icosahedron. The resulting discrete
alphabet is based on a very uniform sampling of $\rm{SE}(3)$ and is a tool for
describing the continuous trajectories of robots and humans. The general
"signals to symbols" problem in artificial intelligence is cast in this
framework for robots moving continuously in the world, and we present a
coarse-to-fine search scheme here to efficiently solve this decoding problem in
practice.