Obtaining Optimal Mobile-Robot Paths with Nonsmooth Anisotropic Cost Functions Using Qualitative-State Reasoning

Abstract

Realistic path-planning problems frequently show anisotropism, dependency of traversal cost or feasibility on the traversal heading. Gravity, friction, visibility, and safety are often anisotropic for mobile robots. Anisotropism often differs qualitatively with heading, as when a vehicle has insufficient power to go uphill or must brake to avoid accelerating down hill. Modeling qualitative distinctions requires discontinuities in either the cost-per-traversal-distance function or its derivatives, preventing direct application of most results of the calculus of variations. We present a new approach to optimal anisotropic path planning that first identifies qualitative states and per missible transitions between them. If the qualitative states are chosen appropriately, our approach replaces an optimization problem with such discontinuities by a set of subproblems with out discontinuities, subproblems for which optimization is likely to be faster and less troublesome. Then the state space in the near neighborhood of any particular state can be partitioned into "behavioral regions" representing states optimally reach able by single qualitative "behaviors, " sequences of qualitative states in a finite-state diagram. Simplification of inequalities and other methods can identify the behavioral regions. Our ideas solve problems that are not easily solvable any other way, especially problems with what we define as "turn-hostile" anisotropism. We illustrate our methods on two examples, navigation on an arbitrarily curved surface with gravity and friction effects (for which we show much better performance than a previously published program 22 times longer), and flight of a simple missile.