Dynamic Navigation in Direction-Dependent Environments

This chapter examines optimal path finding problems with direction, location and time dependent environments. The dependence of the cost function and path constraints on the location of the mobile agent and time creates the need for a dynamic navigation algorithm, capable of adjusting the path in real-time as more information about the environment becomes available. In addition, the direction-dependent nature of the environment results in an asymmetric cost function, which is not a metric and prohibits the use of more traditional and established approaches to solving optimal path finding problems. Moreover, the triangle inequality is often violated for the direction-dependent cost functions, further preventing the use of analysis and results developed for Euclidian shortest path problems. To add another dimension of reality to our model, we integrate the system dynamics and constrain feasible paths by maximum sharpness of a turn that a mobile agent can make. The presented work delivers a more realistic optimal path-finding model while reducing the computational time required to find such a path. This is particularly important since real-time implementation is essential for our applications. In addition, many analytical results derived here provide insights into the structure of the problem, its objective function, and the optimal solution. These insights provide a closed-form solution to a large subset of problems where additional assumptions are applicable. For such problems, we easily construct the analytical solutions instead of implementing more involved, and often approximate, methods presented in the literature. We describe the Optimum Vessel Performance in Evolving Nonlinear Wavefields Project that motivated our work, and deliver computational results to demonstrate the applicability and performance of our path-finding methods. ∗This work was supported in part by the Office of Naval Research through the Multidisciplinary University Research Initiative (MURI) Optimum Vessel Performance in Evolving Nonlinear Wave Fields grant (N0001405-1-0537) and through the Autonomous Vehicle Dynamic Navigation System grant (N00014-11-1-0516).