Many stochastic problems of interest in engineering involve random
rigid-body motions. In this talk, a variety of stochastic phenomena that
evolve on the group of rigid-body motions will be discussed. These
include mobile robot path planning, and robot-arm inverse kinematics.
These topics lead in to our current work on multi-robot team diagnosis
and repair, information fusion, and self-replicating robots. The entropy
of the set of all possible arrangements (or configurations) of spare
parts in the environment is such a measure, and has led us to study
problems at the foundations of statistical mechanics and information
theory. Methods from noncommutative harmonic analysis, probability and
statistics on Lie groups, and hypo-elliptic diffusions arise naturally
in these applications. Similar mathematical issues encountered in
structural biology in the context of DNA statistical mechanics and in
the analysis of protein structure and motions will also be discussed.