Geodesics for Mobile Robots

As mobile robots operate with limited resources in
large obstructed environments, their success is
dependent on how efficiently they move while they
avoid collision with obstacles and other robots.
Therefore, planning optimal motions and devising
optimal coordination strategies are two important and
challenging fundamental problems in mobile robotics,
which have received significant attention in the last
couple of decades. Both of those problems can be
reduced to shortest path, or equivalently geodesic,
problems in appropriate geometric settings. Geodesic
problems have been studied in two disciplines: (1)
optimal control theory, and (2) computational
geometry. Optimal control theory has focused on the
differential constraints of robotic systems, while
computational geometry has focused on shortest path
problems in an environment with obstacles. We
introduce a unified approach that is inspired by main
results in both disciplines. In this book, we
demonstrate our technique, which combines the
celebrated Pontryagin's Maximum Principle from
optimal control theory with visibility graph methods
from computational geometry.