  We report a globally-optimal approach to robotic path planning under
uncertainty, based on the theory of quantitative measures of formal languages.
A significant generalization to the language-measure-theoretic path planning
algorithm $\nustar$ is presented that explicitly accounts for average dynamic
uncertainties and estimation errors in plan execution. The notion of the
navigation automaton is generalized to include probabilistic uncontrollable
transitions, which account for uncertainties by modeling and planning for
probabilistic deviations from the computed policy in the course of execution.
The planning problem is solved by casting it in the form of a performance
maximization problem for probabilistic finite state automata. In essence we
solve the following optimization problem: Compute the navigation policy which
maximizes the probability of reaching the goal, while simultaneously minimizing
the probability of hitting an obstacle. Key novelties of the proposed approach
include the modeling of uncertainties using the concept of uncontrollable
transitions, and the solution of the ensuing optimization problem using a
highly efficient search-free combinatorial approach to maximize quantitative
measures of probabilistic regular languages. Applicability of the algorithm in
various models of robot navigation has been shown with experimental validation
on a two-wheeled mobile robotic platform (SEGWAY RMP 200) in a laboratory
environment.
