Accounting for uncertainty in optimal path planning is essential for many applications. We present and apply stochastic level-set partial differential equations that govern the stochastic time-optimal reachability fronts and time-optimal paths for vehicles navigating in uncertain, strong, and dynamic flow fields. To solve these equations efficiently, we obtain and employ their dynamically orthogonal reduced-order projections, maintaining accuracy while achieving several orders of magnitude in computational speed-up when compared to classic Monte Carlo methods. We utilize the new equations to complete stochastic reachability and time-optimal path planning in three test cases: (i) a canonical stochastic steady-front with uncertain flow strength, (ii) a stochastic barotropic quasi-geostrophic double-gyre circulation, and (iii) a stochastic flow past a circular island. For all the three test cases, we analyze the results with a focus on studying the effect of flow uncertainty on the reachability fronts and time-optimal paths, and their probabilistic properties. With the first test case, we demonstrate the approach and verify the accuracy of our solutions by comparing them with the Monte Carlo solutions.With the second, we show that different flow field realizations can result in paths with high spatial dissimilarity but with similar arrival times. With the third, we provide an example where time-optimal path variability can be very high and sensitive to uncertainty in eddy shedding direction downstream of the island. Keywords: Stochastic Path Planning, Level Set Equations, Dynamically Orthogonal, Ocean Modeling, AUV, Uncertainty Quantification

Deepak Subramani, a sixth year graduate student, has been awarded the SNAME Travel Award in Ocean Engineering by MIT-MechE to present his work at the 2017 American Geophysical Union Fall Meeting, to be held from December 11 to 15, 2017, in New Orleans, Louisiana, U.S.A.

Abstract

The ocean is a prime example of multiscale nonlinear fluid dynamical system. Ocean fields are usually complex, with intermittent features and nonstationary heterogeneous statistics. Due to the limited measurements, there are multiple sources of uncertainties, including the initial conditions, boundary conditions, forcing, parameters, and even the model parameterizations and equations themselves. To reduce uncertainties and allow long-duration measurements, the energy consumption of ocean observing platforms need to be optimized. Predicting the distributions of reachable regions, time-optimal paths, and risk-optimal paths in uncertain, strong and dynamic flows is also essential for their optimal and safe operations. Motivated by the above needs, the objectives of this thesis are to develop and apply the theory, schemes, and computational systems for: (i) Dynamically Orthogonal ocean primitive-equations with a nonlinear free-surface, in order to quantify uncertainties and predict probabilities for four-dimensional (time and 3-d in space) coastal ocean states, respecting their nonlinear governing equations and non-Gaussian statistics; (ii) Stochastic Dynamically Orthogonal level-set optimization for energy-optimal path planning of autonomous agents in coastal regions, rigorously incorporating realistic ocean predictions; (iii) Probabilistic predictions of reachability, time-optimal paths and risk-optimal paths in uncertain, strong and dynamic flows.

The theoretical and computational foundation is laid out and several idealized-to-realistic applications are demonstrated. Examples in the Middle Atlantic Bight region, Northwest Atlantic, and northern Indian ocean are showcased. The probabilistic prediction and path planning methodologies developed here are PDE-based and provide stochastic ocean fields, and energy-optimal, stochastic time-optimal and risk-optimal predictions without heuristics. Computationally, the new methods are several orders of magnitude faster than direct Monte Carlo methods.

Such technologies can be utilized for several commercial and societal applications, now and in the future. Specifically, the probabilistic ocean predictions can be input to a technical decision aide for a sustainable fisheries co-management program in India, which has the potential to provide environment friendly livelihoods to millions of marginal fishermen. The risk-optimal path planning equations can be employed in real-time for efficient ship routing to reduce greenhouse gas emissions and save operational costs.

Arkopal Dutt, a third year graduate student, has been selected to receive a AGU Fall Meeting Student Travel Award to attend the 2017 American Geophysical Union Fall Meeting, to be held from December 11 to 15, 2017, in New Orleans, Louisiana, U.S.A.