Optimal motion planning using navigation measure

We introduce navigation measure as a new tool to solve the motion planning problem in the presence of static obstacles. Existence of navigation measure guarantees collision-free convergence at the final destination set beginning with almost every initial condition with respect to the Lebesgue measure. Navigation measure can be viewed as a dual to the navigation function. While the navigation function has its minimum at the final destination set and peaks at the obstacle set, navigation measure takes the maximum value at the destination set and is zero at the obstacle set. A linear programming formalism is proposed for the construction of navigation measure. Set-oriented numerical methods are utilised to obtain finite dimensional approximation of this navigation measure. Application of the proposed navigation measure-based theoretical and computational framework is demonstrated for a motion planning problem in a complex fluid flow.