The first steps towards integrating autonomous monitoring, probabilistic forecasting, reachability analysis, and adaptive sampling for the Gulf of Mexico were demonstrated in real-time during the collaborative Mini-Adaptive Sampling Test Run (MASTR) ocean experiment, which took place from February to April 2024. The emphasis of this contribution is on the use of the MIT Multidisciplinary Simulation, Estimation, and Assimilation Systems (MSEAS) including Error Subspace Statistical Estimation (ESSE) large-ensemble forecasting and path planning systems to predict ocean fields and uncertainties, forecast reachable sets and optimal paths for gliders, and guide sampling aircraft and ocean vehicles toward the most informative observations. Deterministic and probabilistic ocean forecasts are exemplified and linked to the variability of the Loop Current (LC) and LC Eddies, demonstrating predictive skill by real-time comparisons to independent data. Risk forecasts in terms of probabilities of currents exceeding 1.5 kt were provided. The most informative sampling patterns for Remote Ocean Current Imaging System (ROCIS) flights were forecast using mutual information between surface currents and density anomaly. Finally, we guided four underwater gliders using probabilistic reachability and path-planning forecasts.

As part of the Mini-Adaptive Sampling Test Run (MASTR) experiment in the Gulf of Mexico (GoM) region from February to April 2024, we demonstrated real-time deterministic and probabilistic reachability analysis and time-optimal path planning to guide a fleet of four ocean gliders. The governing differential equations for reachability analysis and time-optimal path planning were numerically integrated in real-time and forced by currents from our large-ensemble ocean forecasts. We illustrate the real-time deterministic and probabilistic forward reachability analyses, reachability and path planning for glider pickups, time-optimal path planning for gliders in distress, and planning of future glider deployments. Results show that the actual paths of gliders were contained within our reachable set forecasts and in accord with the dynamic reachability fronts. Our time-optimal headings and paths also predicted real glider motions, even for longer-range predictions of weeks to a month duration. Reachability and time-optimal path planning forecasts were successfully employed for glider recovery. They also enabled exploring options for future glider deployments.

General differential equations for multi-objective reachability and optimal planning are used to guide autonomous air and sea drones in hazard-time optimal missions. The vehicles minimize exposure to hazards and travel time, leveraging the dynamic environments with strong flows and steering clear of dynamic hazardous regions. We demonstrate the approach first with an autonomous air drone that crosses the Atlantic Ocean optimizing travel time using trade winds while avoiding hazardous rain storms in the inter-tropical convergence zone. We then consider an air drone that exploits winds and avoids hazardous rains to transport an ocean vehicle to a target destination. The ocean vehicle then completes its own hazard-time optimal mission, leveraging ocean currents and avoiding vessel traffic hazards. In all cases, we predict hazard-time reachable sets, Pareto fronts, and optimal paths. The results highlight the benefits of considering hazards in optimal path planning.

Seaweed biomass presents a substantial opportunity for climate mitigation, yet to realize its potential, farming must be expanded to the expansive open oceans. However, in the open ocean neither anchored farming nor floating farms operating with powerful engines are economically viable. Recent studies have shown that vessels can navigate with low-power engines by going with the flow, utilizing minimal propulsion to strategically leverage beneficial ocean currents. In this work, we focus on low-power autonomous seaweed farms and design controllers that maximize seaweed growth by taking advantage of ocean currents. We first introduce a Dynamic Programming (DP) formulation to solve for the growth-optimal value function when the true currents are known. However, in reality only short-term imperfect forecasts with increasing uncertainty are available. Hence, we present three additional extensions. Firstly, we use frequent replanning to mitigate forecast errors. For that we compute the value function daily as new forecasts arrive, which also provides a feedback policy that is equivalent to replanning on the forecast at every time step. Second, to optimize for long-term growth, we extend the value function beyond the forecast horizon by estimating the expected future growth based on seasonal average currents. Lastly, we introduce a discounted finite-time DP formulation to account for the increasing uncertainty in future ocean current estimates. We empirically evaluate our approach with 30-day simulations of farms in realistic ocean conditions. Our method achieves 95.8% of the best possible growth using only 5-day forecasts. This confirms the feasibility of using low-power propulsion to operate autonomous farms in real-world conditions.

Low-propulsion vessels can take advantage of powerful ocean currents to navigate towards a destination. Recent results demonstrated that vessels can reach their destination with high probability despite forecast errors. However, these results do not consider the critical aspect of safety of such vessels: because their propulsion is much smaller than the magnitude of surrounding currents, they might end up in currents that inevitably push them into unsafe areas such as shallow waters, garbage patches, and shipping lanes. In this work, we first investigate the risk of stranding for passively floating vessels in the Northeast Pacific. We find that at least 5.04% would strand within 90 days. Next, we encode the unsafe sets as hard constraints into Hamilton-Jacobi Multi-Time Reachability to synthesize a feedback policy that is equivalent to re-planning at each time step at low computational cost. While applying this policy guarantees safe operation when the currents are known, in realistic situations only imperfect forecasts are available. Hence, we demonstrate the safety of our approach empirically with large-scale realistic simulations of a vessel navigating in high-risk regions in the Northeast Pacific. We find that applying our policy closed-loop with daily re-planning as new forecasts become available reduces stranding below 1% despite forecast errors often exceeding the maximal propulsion. Our method significantly improves safety over the baselines and still achieves a timely arrival of the vessel at the destination.

Long-term and reliable marine ecosystems monitoring is essential to address current environmental issues, including climate change and biodiversity threats. The existing oceans monitoring systems show clear data gaps, particularly when considering characteristics such as depth coverage or measured variables in deep and open seas. Over the last decades, the number of fixed and mobile platforms for in situ ocean data acquisition has increased significantly, covering all oceans’ regions. However, these are largely dependent on satellite communications for data transmission, as well as on research cruises or opportunistic ship surveys, generally presenting a lag between data acquisition and availability. In this context, the creation of a widely distributed network of SMART cables (Science Monitoring And Reliable Telecommunications) – sensors attached to submarine telecommunication cables – appears as a promising solution to fill in the current ocean data gaps and ensure unprecedented oceans health continuous monitoring. The K2D (Knowledge and Data from the Deep to Space) project proposes the development of a persistent oceans monitoring network based on the use of telecommunications cables and Autonomous Underwater Vehicles (AUVs). The approach proposed includes several modules for navigation, communication and energy management, that enable the cost-effective gathering of extensive oceans data. These include physical, chemical, and biological variables, both registered with bottom fixed stations and AUVs operating in the water column. The data that can be gathered have multiple potential applications, including oceans health continuous monitoring and the enhancement of existing ocean models. The latter, in combination with geoinformatics and Artificial Intelligence, can create a continuum from the deep sea to near space, by integrating underwater remote sensing and satellite information to describe Earth systems in a holistic manner.

For the analysis of dynamical systems, it is fundamental to determine all states that can be reached at any given time. In this work, we obtain and apply new governing equations for reachability analysis over multiple start and terminal times all at once, and for systems operating in time-varying environments with dynamic obstacles and any other relevant dynamic fields. The theory and schemes are developed for both backward and forward reachable tubes with time-varying target and start sets. The resulting value functions elegantly capture not only the reachable tubes but also time-to-reach and time-to-leave maps as well as start time vs. duration plots and other useful secondary quantities for optimal control. We discuss the numerical schemes and computational efficiency. We first verify our results in an environment with a moving target and obstacle where reachability tubes can be analytically computed. We then consider the Dubin’s car problem extended with a moving target and obstacle. Finally, we showcase our multi-time reachability in a non-hydrostatic bottom gravity current system. Results highlight the novel capabilities of exact multi-time reachability in dynamic environments.

In dynamic flow fields such as winds and ocean currents an agent can navigate by going with the flow, only using minimal propulsion to nudge itself into beneficial flows. This navigation paradigm of hitchhiking flows is highly energy-efficient. However, reliable navigation in this setting remains challenging as typically only forecasts are available which differ significantly from the true currents and the forecast error can be larger than can be handled by the actuation of the agent. In this paper, we propose a novel control method for reliable navigation of underactuated agents hitchhiking flows based on imperfect forecasts. In the spirit of Model Predictive Control our method allows for time-optimal replanning at every time step with only one computation per forecast. Using the recent Multi-Time Hamilton-Jacobi Reachability formulation we obtain a value function which is then used for closed-loop control. We evaluate the reliability of our method empirically over a large set of multi-day start-target missions in the ocean currents of the Gulf of Mexico with realistic forecast errors. Our method outperforms the baselines significantly, achieving high reliability, measured as the success rate of navigating from start to target, at low computational cost.

The rising popularity of aquaculture has led to increased research in offshore algae farming. Central to the efficient operation of such farms is the need for (i) accurate models of the dynamic ocean environment including macroalgae ecosystem dynamics and (ii) intelligent path planning algorithms for autonomous vessels that optimally manage and harvest the algae fields. In this work, we address both these challenges. We first integrate our modeling system of the ocean environment with a model for forecasting the growth and decay of algae fields. These fields are then input into our exact optimal path planning, augmented with the optimal harvesting goals and solved using level set methods. The resulting path is a provable time-optimal route for the vehicle to follow under the constraint of having to monitor or harvest a specified amount of the field to collect. To demonstrate the theory, we simulate algal growth in both idealized and realistic data-assimilative dynamic ocean environments and compute the optimal paths for an autonomous collection vehicle. We demonstrate that our theory and schemes can be used to compute the optimal path in a variety of scenarios – harvesting in the case of discrete farms, a large kelp farm field, or large scale dynamic algal bloom fields.

The present paper demonstrates the use of exact equations to predict time-optimal mission plans for a marine vehicle that visits a number of locations in a given dynamic ocean current field. This problem bears close resemblance to that of the classic “traveling salesman”, albeit with the added complexity that the vehicle experiences a dynamic flow field while traversing the paths. The paths, or “legs”, between all goal waypoints are generated by numerically solving the exact time-optimal path planning level-set differential equations. Overall, the planning proceeds in four steps. First, current forecasts for the planning horizon is obtained utilizing our data-driven 4-D primitive equation ocean modeling system (Multidisciplinary Simulation Estimation and Assimilation System; MSEAS), forced by high-resolution tidal and real-time atmopsheric forcing fields. Second, all tour permutations are enumerated and the minimum number of times the time-optimal PDEs are to be solved is established. Third, due to the spatial and temporal dynamics, a varying start time results in different paths and durations for each leg and requires all permutations of travel to be calculated. To do so, the minimum required time-optimal PDEs are solved and the optimal travel time is computed for each leg of all enumerated tours. Finally, the tour permutation for which travel time is minimized is identified and the corresponding time-optimal paths are computed by solving the backtracking equation. Even though the method is very efficient and the optimal path can be computed serially in real-time for common naval operations, for additional computational speed, a high-performance computing cluster was used to solve the level set calculations in parallel. Our equation and software is applied to simulations of realistic naval applications in the complex Philippines Archipelago region. Our method calculates the global optimum and can serve two purposes: (a) it can be used in its present form to plan multiwaypoint missions offline in conjunction with a predictive ocean current modeling system, or (b) it can be used as a litmus test for approximate future solutions to the traveling salesman problem in dynamic flow fields.

Recent advances in probabilistic forecasting of regional ocean dynamics, and stochastic optimal path planning with massive ensembles motivate principled analysis of their large datasets. Specifically, stochastic time-optimal path planning in strong, dynamic and uncertain ocean flows produces a massive dataset of the stochastic distribution of exact timeoptimal trajectories. To synthesize such big data and draw insights, we apply machine learning and data mining algorithms. Particularly, clustering of the time-optimal trajectories is important to describe their PDFs, identify representative paths, and compute and optimize risk of following these paths. In the present paper, we explore the use of hierarchical clustering algorithms along with a dissimilarity matrix computed from the pairwise discrete Frechet distance between all the optimal trajectories. We apply the algorithms to two datasets of massive ensembles of vehicle trajectories in a stochastic flow past a circular island and stochastic wind driven double gyre flow. These paths are computed by solving our dynamically orthogonal level set equations. Hierarchical clustering is applied to the two datasets, and results are qualitatively and quantitatively analyzed.

In recent years, the use of autonomous undersea vehicles (AUVs) for highly time-critical at-sea operations involving surface ships has received increased attention, magnifying the importance of optimal interception. Finding the optimal route to a moving target is a challenging procedure. In this work, we describe and apply our exact time-optimal path planning methodology and the corresponding software to such ship interception problems. A series of numerical ship interception experiments is completed in the southern littoral of Massachusetts, namely in Buzzards Bay and Vineyard Sound around the Elizabeth Islands and Martha’s Vineyard. Ocean currents are estimated from a regional ocean modeling system. We show that complex coastal geometry, ship proximity, and tidal current phases all play key roles influencing the time-optimal vehicle behavior. Favorable or adverse currents can shift the optimal route from one island passage to another, and can even cause the AUV to remain nearly stationary until a favorable current develops. We also integrate the Kelvin wedge wake model into our path planning software, and show that considering wake effects significantly complicates the shape of the time-optimal paths, requiring AUVs to execute sequences of abrupt turns and tacking maneuvers, even in highly idealized scenarios. Such behavior is reminiscent of ocean animals swimming in wakes. In all cases, it is shown that our level set partial differential equations successfully guide the time-optimal vehicles through regions with the most favorable currents, avoiding regions with adverse effects, and accounting for the ship wakes when present.

Autonomous underwater vehicles (AUVs) are used to execute an increasingly challenging set of missions in commercial, environmental and defense industries. The resources available to the AUV in service of these missions are typically a limited power supply and onboard sensing of its local environment. Optimal path planning is needed to maximize the chances that these AUVs will successfully complete long endurance missions within their power budget. A time-optimal path planner has been recently developed to minimize AUV mission time required to traverse a dynamic ocean environment at a specified speed through the water. For many missions, time minimization is appropriate because the AUVs operate at a fixed propeller speed. However, the ultimate limiting constraint on AUV operations is often the onboard power supply, rather than mission time. While an empirical or theoretical relationship between mission time and power could be applied to estimate power usage in the path planner, the real power usage and availability on an AUV varies mission-to-mission, as a result of multiple factors, including vehicle buoyancy, battery charge cycle, fin configuration, and water type or quality. In this work, we use data collected from two mid-size AUVs operating in various conditions to learn the mission-to-mission variability in the power budget so that it could be incorporated into the mission planner.

We report the results of sea exercises that demonstrate the real-time capabilities of our fundamental time-optimal path planning theory and software with real ocean vehicles. The exercises were conducted with REMUS 600 Autonomous Underwater Vehicles (AUVs) in the Buzzards Bay and Vineyard Sound Regions on 21 October and 6 December 2016. Two tests were completed: (i) 1-AUV time-optimal tests and (ii) 2-AUV race tests where one AUV followed a time-optimal path and the other a shortest-distance path between the start and finish locations. The time-optimal planning proceeded as follows. We first forecast, in real-time, the physical ocean conditions in the above regions and times utilizing our MSEAS multi-resolution primitive equation ocean modeling system. Next, we planned time-optimal paths for the AUVs using our level-set equations and real-time ocean forecasts, and accounting for operational constraints (e.g. minimum depth). This completed the planning computations performed onboard a research vessel. The forecast optimal paths were then transferred to the AUV operating system and the vehicles were piloted according to the plan. We found that the forecast currents and paths were accurate. In particular, the time-optimal vehicles won the races, even though the local currents and geometric constraints were complex. The details of the results were analyzed off-line after the sea tests.

In this paper, we adopt a reachability-based approach to deal with the pursuit-evasion differential game between two players in the presence of dynamic environmental disturbances (e.g., winds, sea currents). We give conditions for the game to be terminated in terms of reachable set inclusions. Level set equations are defined and solved to generate the reachable sets of the pursuer and the evader. The corresponding time-optimal trajectories and optimal strategies can be retrieved immediately afterwards. We validate our method by applying it to a pursuit-evasion game in a simple flow field, for which an analytical solution is available.We then implement the proposed scheme to a problem with a more realistic flow field.

We present a novel stochastic optimization method to compute energy-optimal paths, among all time-optimal paths, for vehicles traveling in dynamic unsteady currents. The method defines a stochastic class of instantaneous nominal vehicle speeds and then obtains the energy-optimal paths within the class by minimizing the total time-integrated energy usage while still satisfying the strong-constraint time-optimal level set equation. This resulting stochastic level set equation is solved using a dynamically orthogonal decomposition and the energy-optimal paths are then selected for each arrival time, among all stochastic time-optimal paths. The first application computes energy-optimal paths for crossing a steady front. Results are validated using a semi-analytical solution obtained by solving a dual nonlinear energy-time optimization problem. The second application computes energy-optimal paths for a realistic mission in the Middle Atlantic Bight and New Jersey Shelf/Hudson Canyon region, using dynamic data-driven ocean field estimates.

We develop and illustrate an efficient but rigorous methodology that predicts the time-optimal paths of ocean vehicles in dynamic continuous flows. The goal is to best utilize or avoid currents, without limitation on these currents nor on the number of vehicles. The methodology employs a new modified level set equation to evolve a wavefront from the starting point of vehicles until they reach their desired goal locations, combining flow advection with nominal vehicle motions. The optimal paths of vehicles are then computed by solving particle tracking equations backwards in time. The computational cost is linear with the number of vehicles and geometric with spatial dimensions. The methodology is applicable to any continuous flows and many vehicles scenarios. Present illustrations consist of the crossing of a canonical uniform jet and its validation with an optimization problem, as well as more complex time varying 2D flow fields, including jets, eddies and forbidden regions.