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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.

Pereira, E., M. Tieppo, J. Faria, D. Hart, P. Lermusiaux, and the K2D Project Team, 2023. *Subsea Cables as Enablers of a Next Generation Global Ocean Sensing System*. Oceanography 36(Supplement 1). doi:10.5670/oceanog.2023.s1.22. Special issue: "Frontiers in Ocean Observing: Emerging Technologies for Understanding and Managing a Changing Ocean"

The ocean is vast, complex, and increasingly threatened by human activities. There is an urgent need to find complementary ways to gather information and promote the comprehensive understanding and management of the ocean. The global network of subsea cables provides an opportunity to support a holistic ocean observation system. Data gathered from this system can be employed to anticipate and provide warning about hazardous events. Large-scale and widespread ocean monitoring may also enable the oversight and tracing of global phenomena that have local impacts.

The Knowledge and Data from the Deep to Space (K2D) project aims to develop the critical components that will enable the large-scale coupling of autonomous underwater vehicles (AUVs) and subsea cables for global ocean environmental monitoring and multi-hazard warning. Funded by the Fundação para a Ciência e Tecnologia/Massachusetts Institute of Technology Portugal Program and involving teams from Portugal and the United States, the project started in 2021 with a global budget of 1.4 M€ and an estimated duration of three years. Sustained ocean observation systems are scarce, especially those that focus on or near the seafloor. The combination of subsea cables and marine robotics is promising not only because it allows access to remote locations and provides an extensive network (deep sea, open ocean), but also because it combines a large set of capabilities in a highly resource-efficient way, unmatched by any other ocean observation approach. These assets may initiate the first global ocean “nervous system” in the near future.

Doshi, M.M., M.S. Bhabra, and P.F.J. Lermusiaux, 2023. *Energy-Time Optimal Path Planning in Dynamic Flows: Theory and Schemes*. Computer Methods in Applied Mechanics and Engineering 405: 115865. doi:10.1016/j.cma.2022.115865

We obtain, solve, and verify fundamental differential equations for energy-time path planning in dynamic flows. The equations govern the energy-time reachable sets, optimal paths, and optimal controls for autonomous vehicles navigating to any destination in known dynamic environments, minimizing both energy usage and travel time. Based on Hamilton-Jacobi theory for reachability and the level set method, the resulting methodology computes the Pareto optimal solutions to the multi-objective path planning problem, numerically solving the exact equations governing the evolution of reachability fronts and optimal paths in the augmented energy and physical-space domain. Our approach is applicable to path planning in various dynamic flow environments and energy types. We first validate the methodology through a benchmark case of crossing a steady jet for which we compare our results to semi-analytical optimal energy-time solutions. We then consider unsteady flow environments and solve for energy-time optimal missions in a quasi-geostrophic double-gyre flow field. Results show that our theory and schemes can provide all the energy-time optimal solutions and that these solutions can be strongly influenced by unsteady flow conditions.

Cococcioni, M., L. Fiaschi, and P.F.J. Lermusiaux, 2021. *Game Theory for Unmanned Vehicles Path Planning in the Marine Domain: State of the Art and New Possibilities*. Journal of Marine Science and Engineering 9(11), 1175. doi:10.3390/jmse9111175. Special Issue on Machine Learning and Remote Sensing in Ocean Science and Engineering.

Thanks to the advent of new technologies and higher real-time computational capabilities, the use of unmanned vehicles in the marine domain received a significant burst in the last decade. Ocean and seabed sampling, missions in dangerous areas, and civilians security are just a few of the large number of applications which currently benefit from unmanned vehicles. One of the most actively studied topic is their full autonomy, i.e., the design of marine vehicles capable of pursuing a task while reacting to the changes of the environment without the intervention of humans, not even remote. Environment dynamicity may consist in variations of currents, presence of unknown obstacles, and attacks from adversaries (e.g., pirates). To achieve autonomy in such highly dynamic uncertain conditions, many types of autonomous path planning problems need to be solved. There has thus been a commensurate number of approaches and methods to optimize such path planning. This work focuses on *game theoretic* ones and provides a wide overview of the current state of the art, along with future directions.

Mannarini, G., D.N. Subramani, P.F.J. Lermusiaux, and N. Pinardi, 2020. *Graph-Search and Differential Equations for Time-Optimal Vessel Route Planning in Dynamic Ocean Waves*, IEEE Transactions on Intelligent Transportation Systems 21(8), 3581-3593, doi:10.1109/TITS.2019.2935614

Time-optimal paths are evaluated by VISIR (“discoVerIng Safe and effIcient Routes”), a graph-search ship routing model, with respect to the solution of the fundamental differential equations governing optimal paths in a dynamic wind-wave environment. The evaluation exercise makes use of identical setups: topological constraints, dynamic wave environmental conditions, and vessel-ocean parametrizations, while advection by external currents is not considered. The emphasis is on predicting the time-optimal ship headings and Speeds Through Water constrained by dynamic ocean wave fields. VISIR upgrades regarding angular resolution, time-interpolation, and static navigational safety constraints are introduced. The deviations of the graph-search results relative to the solution of the exact differential equations in both the path duration and length are assessed. They are found to be of the order of the discretization errors, with VISIR’s solution converging to that of the differential equation for sufficient resolution.

Kulkarni, C.S. and P.F.J. Lermusiaux, 2020. *Three-Dimensional Time-Optimal Path Planning in the Ocean*, Ocean Modelling, 152, 101644. doi:10.1016/j.ocemod.2020.101644

Autonomous underwater vehicles (AUVs) operate in the three-dimensional and time-dependent marine environment with strong and dynamic currents. Our goal is to predict the time history of the optimal three-dimensional headings of these vehicles such that they reach the given destination location in the least amount of time, starting from a known initial position. We employ the exact differential equations for time-optimal path planning and develop theory and numerical schemes to accurately predict three-dimensional optimal paths for several classes of marine vehicles, respecting their specific propulsion constraints. We further show that the three-dimensional path planning problem can be reduced to a two-dimensional one if the motion of the vehicle is partially known, e.g. if the vertical component of the motion is forced. This reduces the computational cost. We then apply the developed theory in three-dimensional analytically known flow fields to verify the schemes, benchmark the accuracy, and demonstrate capabilities. Finally, we showcase time-optimal path planning in realistic data-assimilative ocean simulations for the Middle Atlantic Bight region, integrating the primitive-equation of the Multidisciplinary Simulation Estimation and Assimilation System (MSEAS) with the three-dimensional path planning equations for three common marine vehicles, namely propelled AUVs (with unrestricted motion), floats (that only propel vertically), and gliders (that often perform sinusoidal yo-yo motions in vertical planes). These results highlight the effects of dynamic three-dimensional multiscale ocean currents on the optimal paths, including the Gulf Stream, shelfbreak front jet, upper-layer jets, eddies, and wind-driven and tidal currents. They also showcase the need to utilize data-assimilative ocean forecasts for planning efficient autonomous missions, from optimal deployment and pick-up, to monitoring and adaptive data collection.

Subramani, D.N. and P.F.J. Lermusiaux, 2019. *Risk-Optimal Path Planning in Stochastic Dynamic Environments*. Computer Methods in Applied Mechanics and Engineering, 353, 391–415. doi:10.1016/j.cma.2019.04.033

We combine decision theory with fundamental stochastic time-optimal path planning to develop partial-differential-equations-based schemes for risk-optimal path planning in uncertain, strong and dynamic flows. The path planning proceeds in three steps: (i) predict the probability distribution of environmental flows, (ii) compute the distribution of exact time-optimal paths for the above flow distribution by solving stochastic dynamically orthogonal level set equations, and (iii) compute the risk of being suboptimal given the uncertain time-optimal path predictions and determine the plan that minimizes the risk. We showcase our theory and schemes by planning risk-optimal paths of unmanned and/or autonomous vehicles in illustrative idealized canonical flow scenarios commonly encountered in the coastal oceans and urban environments. The step-by-step procedure for computing the risk-optimal paths is presented and the key properties of the risk-optimal paths are analyzed.

Moore, A.M., M. Martin, S. Akella, H. Arango, M. Balmaseda, L. Bertino, S. Ciavatta, B. Cornuelle, J. Cummings, S. Frolov,
P. Lermusiaux, P. Oddo, P.R. Oke, A. Storto, A. Teruzzi, A. Vidard, and A.T. Weaver, 2019. *Synthesis of Ocean Observations using Data
Assimilation for Operational, Real-time and Reanalysis Systems: A More Complete Picture of the State of the Ocean*. Frontiers
in Marine Science 6(90), 1–6. doi:10.3389/fmars.2019.00090

Gil, Y., S.A. Pierce, H. Babaie, A. Banerjee, K. Borne, G. Bust, M. Cheatham, I. Ebert-Uphoff, C. Gomes, M. Hill, J. Horel, L. Hsu, J. Kinter, C. Knoblock, D. Krum, V. Kumar, P.F.J. Lermusiaux, Y. Liu, C. North, V. Pankratius, S. Peters, B. Plale, A. Pope, S. Ravela, J. Restrepo, A. Ridley, H. Samet, and S. Shekhar, 2019. *Intelligent Systems for Geosciences: An Essential Research Agenda*. Communications of the ACM, 62(1), 76–84. doi:10.1145/3192335

Many aspects of geosciences pose novel problems for intelligent systems research. Geoscience data is challenging because it tends to be uncertain, intermittent, sparse, multiresolution, and multiscale. Geosciences processes and objects often have amorphous spatiotemporal boundaries. The lack of ground truth makes model evaluation, testing, and comparison difficult. Overcoming these challenges requires breakthroughs that would significantly transform intelligent systems, while greatly benefitting the geosciences in turn. Although there have been significant and beneficial interactions between the intelligent systems and geosciences communities, the potential for synergistic research in intelligent systems for geosciences is largely untapped. A recently launched Research Coordination Network on Intelligent Systems for Geosciences followed a workshop at the National Science Foundation on this topic. This expanding network builds on the momentum of the NSF EarthCube initiative for geosciences, and is driven by practical problems in Earth, ocean, atmospheric, polar, and geospace sciences. Based on discussions and activities within this network, this article presents a research agenda for intelligent systems inspired by geosciences challenges.

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

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, J. Marine Res. 75(6), pp. 765–813. https://doi.org/10.1357/002224017823524035

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.

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.

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.

Yilmaz, N.K., C. Evangelinos, N.M. Patrikalakis, P.F.J. Lermusiaux, P.J. Haley, W.G. Leslie, A.R. Robinson, D. Wang and H. Schmidt, 2006a. *Path Planning Methods for Adaptive Sampling of Environmental and Acoustical Ocean Fields*, Oceans 2006, 6pp, Boston, MA, 18-21 Sept. 2006, doi: 10.1109/OCEANS.2006.306841.