Energy optimal path planning by stochastic dynamically orthogonal level set optimization
Abstract A stochastic optimization methodology is formulated for computing energy optimal paths from among time optimal paths of autonomous vehicles navigating in a dynamic flow field. Based on partial differential equations, the methodology rigorously leverages the level set equation that governs time optimal reachability fronts for a given relative vehicle speed function. To set up the energy optimization, the relative vehicle speed and headings are considered to be stochastic and new stochastic Dynamically Orthogonal (DO) level set equations are derived. Their solution provides the distribution of time optimal reachability fronts and corresponding distribution of time optimal paths. An optimization is then performed on the vehicle's energy time joint distribution to select the energy optimal paths for each arrival time, among all stochastic time optimal paths for that arrival time. Numerical schemes to solve the reduced stochastic DO level set equations are obtained, and accuracy and efficiency considerations are discussed. These reduced equations are first shown to be efficient at solving the governing stochastic level sets, in part by comparisons with direct Monte Carlo simulations. To validate the methodology and illustrate its accuracy, comparisons with semi analytical energy optimal path solutions are then completed. In particular, we consider the energy optimal crossing of a canonical steady front and set up its semi analytical solution using a energy time nested nonlinear double optimization scheme. We then showcase the inner workings and nuances of the energy optimal path planning, considering different mission scenarios. Finally, we study and discuss results of energy optimal missions in a wind driven barotropic quasi geostrophic double gyre ocean circulation.