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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

A theoretical synthesis of forward reachability for minimum–time control of anisotropic vehicles operating in strong and dynamic flows is provided. The synthesis relies on the computation of the forward reachable set of states. Using ideas rooted in the theory of non–smooth calculus, we prove that this set is governed by the viscosity solution of an unsteady Hamilton–Jacobi (HJ) equation. We show that the minimum arrival time satisfies a static HJ equation, when a special local controllability condition holds. Results are exemplified by applications to a sailboat moving in a uniform wind–field and autonomous underwater gliders operating in the Sulu Archipelago.

Lermusiaux, P.F.J., D.N. Subramani, J. Lin, C.S. Kulkarni, A. Gupta,
A. Dutt, T. Lolla, P.J. Haley Jr., W.H. Ali, C. Mirabito, and S. Jana, 2017. *A Future for Intelligent Autonomous Ocean Observing Systems.* The Sea. Volume 17, The Science of Ocean Prediction, Part 2, Special Issue, J. Marine Res. 75. In press.

Centurioni, L.R., V. Hormann, L. D. Talley, I. Arzeno, L. Beal, M. Caruso, P. Conry, R. Echols, H. J. S. Fernando, S. N. Giddings, A. Gordon, H. Graber, R. Harcourt, S. R. Jayne, T. G. Jensen, C. M. Lee, P. F. J. Lermusiaux, P. L’Hegaret, A. J. Lucas, A. Mahadevan, J. L. McClean, G. Pawlak, L. Rainville, S. Riser, H. Seo, A. Y. Shcherbina, E. Skyllingstad, J. Sprintall, B. Subrahmanyam, E. Terrill, R. E. Todd, C. Trott, H. N. Ulloa, and H. Wang, 2017. *Northern Arabian Sea Circulation-Autonomous Research (NASCar): A Research Initiative Based on Autonomous Sensors*. Oceanography 30(2):74–87, https://doi.org/10.5670/oceanog.2017.224.

Lermusiaux, P.F.J., P.J. Haley Jr., S. Jana, A. Gupta, C.S. Kulkarni, C. Mirabito,
W.H. Ali, D.N. Subramani, A. Dutt, J. Lin, A. Y. Shcherbina, C. M. Lee, and A. Gangopadhyay, 2017. *Optimal Planning and Sampling Predictions for Autonomous and Lagrangian Platforms and Sensors in the Northern Arabian Sea*. Oceanography 30(2):172–185, https://doi.org/10.5670/oceanog.2017.242.

We integrate data-driven ocean modeling with the stochastic Dynamically
Orthogonal (DO) level-set optimization methodology to compute and study energy-optimal
paths, speeds, and headings for ocean vehicles in the Middle-Atlantic Bight (MAB) region.
We hindcast the energy-optimal paths from among exact time-optimal paths for
the period 28 August 2006 to 9 September 2006. To do so, we first obtain a data-assimilative
multiscale re-analysis, combining ocean observations with implicit two-way nested multiresolution
primitive-equation simulations of the tidal-to-mesoscale dynamics in the region.
Second, we solve the reduced-order stochastic DO level-set partial differential equations
(PDEs) to compute the joint probability of minimum arrival-time, vehicle-speed
time-series, and total energy utilized. Third, for each arrival time, we select the vehiclespeed
time-series that minimize the total energy utilization from the marginal probability
of vehicle-speed and total energy. The corresponding energy-optimal path and headings
are obtained through a particle backtracking equation. Theoretically, the present
methodology is PDE-based and provides fundamental energy-optimal predictions without
heuristics. Computationally, it is three- to four-orders of magnitude faster than direct
Monte Carlo methods. For the missions considered, we analyze the effects of the regional
tidal currents, strong wind events, coastal jets, shelfbreak front, and other local
circulations on the energy-optimal paths. Results showcase the opportunities for vehicles
that intelligently utilize the ocean environment to minimize energy usage, rigorously
integrating ocean forecasting with optimal control of autonomous vehicles.

In this paper a reachability-based approach is adopted to deal with the
pursuit-evasion dierential game between one evader and multiple pursuers
in the presence of dynamic environmental disturbances (e.g., winds, sea
currents). Conditions for the game to be terminated are given in terms
of reachable set inclusions. Level set equations are defined and solved to
generate the forward reachable sets of the pursuers and the evader. The
time-optimal trajectories and the corresponding optimal strategies are sub-
sequently retrieved from these level sets. The pursuers are divided into
active pursuers, guards, and redundant pursuers according to their respec-
tive roles in the pursuit-evasion game. The proposed scheme is implemented on problems with both simple and realistic time-dependent flow fields, with and without obstacles.

Regional ocean models are capable of forecasting conditions for usefully long intervals of time
(days) provided that initial and ongoing conditions can be measured. In resource-limited circumstances, the
placement of sensors in optimal locations is essential. Here, a nonlinear optimization approach to determine
optimal adaptive sampling that uses the Genetic Algorithm (GA) method is presented. The method determines
sampling strategies that minimize a user-defined physics-based cost function. The method is evaluated using
identical twin experiments, comparing hindcasts from an ensemble of simulations that assimilate data selected
using the GA adaptive sampling and other methods. For skill metrics, we employ the reduction of the
ensemble root-mean-square-error (RMSE) between the “true” data-assimilative ocean simulation and the
different ensembles of data-assimilative hindcasts. A 5-glider optimal sampling study is set up for a 400 km x
400 km domain in the Middle Atlantic Bight region, along the New Jersey shelf-break. Results are compared
for several ocean and atmospheric forcing conditions.

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 is 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 overall 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 dual energy–time nested nonlinear 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 strong dynamic double–gyre flow field.

Lermusiaux P.F.J, T. Lolla, P.J. Haley. Jr., K. Yigit, M.P. Ueckermann, T. Sondergaard and W.G. Leslie, 2016. *Science of Autonomy: Time-Optimal Path Planning and Adaptive Sampling for Swarms of Ocean Vehicles*. Chapter 21, Springer Handbook of Ocean Engineering: Autonomous Ocean Vehicles, Subsystems and Control, Tom Curtin (Ed.), pp. 481-498. doi:10.1007/978-3-319-16649-0_21.

Lolla, T., P.J. Haley. Jr. and P.F.J. Lermusiaux, 2015. *Path Planning in Multi-scale Ocean Flows: Coordination and Dynamic Obstacles*. Ocean Modelling, 94, 46-66. DOI: 10.1016/j.ocemod.2015.07.013.

As the concurrent use of multiple autonomous vehicles in ocean missions grows, systematic control for their coordinated operation is becoming a necessity. Many ocean vehicles, especially those used in longer–range missions, possess limited operating speeds and are thus sensitive to ocean currents. Yet, the effect of currents on their trajectories is ignored by many coordination techniques. To address this issue, we first derive a rigorous level-set methodology for distance–based coordination of vehicles operating in minimum time within strong and dynamic ocean currents. The new methodology integrates ocean modeling, time-optimal level-sets and optimization schemes to predict the ocean currents, the short-term reachability sets, and the optimal headings for the desired coordination. Schemes are developed for dynamic formation control, where multiple vehicles achieve and maintain a given geometric pattern as they carry out their missions. Secondly, we obtain an efficient, non–intrusive technique for level-set-based time–optimal path planning in the presence of moving obstacles. The results are time-optimal path forecasts that rigorously avoid moving obstacles and sustain the desired coordination. They are exemplified and investigated for a variety of simulated ocean flows. A wind–driven double–gyre flow is used to study time-optimal dynamic formation control. Currents exiting an idealized strait or estuary are employed to explore dynamic obstacle avoidance. Finally, results are analyzed for the complex geometry and multi–scale ocean flows of the Philippine Archipelago.

The level set methodology for time-optimal path planning is employed to predict collision-free and fastest time trajectories for swarms of underwater vehicles deployed in the Philippine Archipelago region.
To simulate the multiscale ocean flows in this complex region, a data-assimilative primitive-equation ocean modeling system is employed with
telescoping domains that are interconnected by implicit two-way nesting.
These data-driven multiresolution simulations provide a
realistic flow environment, including variable large-scale currents,
strong jets, eddies, wind-driven currents and tides.
The properties and capabilities of the rigorous level set methodology are
illustrated and assessed quantitatively for several vehicle types and mission scenarios.
Feasibility studies of all-to-all broadcast missions, leading to minimal time transmission between source and receiver locations, are performed using a large number of vehicles.
The results with gliders and faster propelled vehicles are compared.
Reachability studies, i.e.~determining the boundaries of regions that can be reached by vehicles for exploratory missions, are then exemplified and analyzed.
Finally, the methodology is used to determine the optimal strategies
for fastest time pick-up of deployed gliders by means of
underway surface vessels or stationary platforms.
The results highlight the complex effects of multiscale flows on the optimal paths,
the need to utilize the ocean environment for more efficient autonomous
missions and the benefits of including ocean forecasts in the planning of time-optimal paths.

We develop an accurate partial differential equation based methodology that predicts the time-optimal paths of autonomous vehicles navigating in any continuous, strong and dynamic ocean currents, obviating the need for heuristics. The goal is to predict a sequence of steering directions so that vehicles can best utilize or avoid currents to minimize their travel time. Inspired by the level set method, we derive and demonstrate that a modified level set equation governs the time-optimal path in any continuous flow. We show that our algorithm is computationally efficient and apply it to a number of experiments. First, we validate our approach through a simple benchmark application in a Rankine vortex flow for which an analytical solution is available. Next, we apply our methodology to more complex, simulated flow-fields such as unsteady double-gyre flows driven by wind stress and flows behind a circular island. These examples show that time-optimal paths for multiple vehicles can be planned, even in the presence of complex flows in domains with obstacles. Finally, we present, and support through illustrations, several remarks that describe specific features of our methodology.

Wang, D., P.F.J. Lermusiaux, P.J. Haley, D. Eickstedt, W.G. Leslie and H. Schmidt, 2009. *Acoustically Focused Adaptive Sampling and On-board Routing for Marine Rapid Environmental Assessment.* Special issue of Journal of Marine Systems on "Coastal processes: challenges for monitoring and prediction", Drs. J.W. Book, Prof. M. Orlic and Michel Rixen (Guest Eds), 78, S393-S407, doi: 10.1016/j.jmarsys.2009.01.037.

The goal of adaptive sampling in the ocean is to predict
the types and locations of additional ocean measurements that
would be most useful to collect. Quantitatively, what is most useful
is defined by an objective function and the goal is then to optimize
this objective under the constraints of the available observing network.
Examples of objectives are better oceanic understanding, to
improve forecast quality, or to sample regions of high interest. This
work provides a new path-planning scheme for the adaptive sampling
problem. We define the path-planning problem in terms of
an optimization framework and propose a method based on mixed
integer linear programming (MILP). The mathematical goal is to
find the vehicle path that maximizes the line integral of the uncertainty
of field estimates along this path. Sampling this path can improve
the accuracy of the field estimates the most. While achieving
this objective, several constraints must be satisfied and are implemented.
They relate to vehicle motion, intervehicle coordination,
communication, collision avoidance, etc. The MILP formulation is
quite powerful to handle different problem constraints and flexible
enough to allow easy extensions of the problem. The formulation
covers single- and multiple-vehicle cases as well as singleand
multiple-day formulations. The need for a multiple-day formulation
arises when the ocean sampling mission is optimized for
several days ahead. We first introduce the details of the formulation,
then elaborate on the objective function and constraints, and
finally, present a varied set of examples to illustrate the applicability
of the proposed method.

Lermusiaux, P.F.J, 2007. *Adaptive Modeling, Adaptive Data Assimilation and Adaptive Sampling.* Refereed invited manuscript. Special issue on "Mathematical Issues and Challenges in Data Assimilation for Geophysical Systems: Interdisciplinary Perspectives". C.K.R.T. Jones and K. Ide, Eds. Physica D, Vol 230, 172-196, doi:
10.1016/j.physd.2007.02.014.

The problem of how to optimally deploy a suite of sensors to estimate the oceanographic
environment is addressed. An optimal way to estimate (nowcast) and predict (forecast)
the ocean environment is to assimilate measurements from dynamic and uncertain regions
into a dynamical ocean model. In order to determine the sensor deployment strategy
that optimally samples the regions of uncertainty, a Genetic Algorithm (GA) approach
is presented. The scalar cost function is defined as a weighted combination of a sensor
suite’s sampling of the ocean variability, ocean dynamics, transmission loss sensitivity,
modeled temperature uncertainty (and others). The benefit of the GA approach is that the
user can determine “optimal” via a weighting of constituent cost functions, which can
include ocean dynamics, acoustics, cost, time, etc. A numerical example with three gliders,
two powered AUVs, and three moorings is presented to illustrate the optimization
approach in the complex shelfbreak region south of New England.