Deepak N. Subramani

A Stochastic Optimization Method for Energy-Based Path Planning

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.

On the Effect of Non-Raining Parameters in Retrieval of Surface Rain Rate Using TRMM PR and TMI Measurements

Microwave radiation with the inherent advantage of its ability to partially penetrate clouds is ideally suited for remote measurements of precipitation, especially over the oceanic regions. The retrieval problem is of great practical interest, as precipitation over the oceans has to be necessarily remotely sensed. More importantly, precipitation is also a crucial input in many weather and climate models. A downward looking space borne radiometer such as the TRMMs Microwave Imager (TMI), however, does not sense the surface rain fall directly. Rather, it measures the upwelling radiation coming from the top-of-atmosphere which depends on the total quantities of the parameters that can affect the radiation in the chosen frequency range. In addition to precipitation, the attenuation and augmentation of total cloud content and integrated precipitable water content by absorption and emission, respectively, affect the microwave brightness temperatures in various frequencies. In the present work, a systematic study has been conducted to investigate the effect of total cloud and precipitable water contents on surface rainrate retrievals from the TMI measured brightness temperatures (BT). While a Bayesian framework is used to assimilate radar reflectivities into hydrometeor structures, a neural network is used to correlate rain and cloud parameters with brightness temperatures. The correlation between the TMI brightness temperatures and the TRMMs Precipitation Radar (TRMM-PR) is used as the benchmark for comparison. A community developed software meso-scale Weather Research and Forecast (WRF) is used to simulate the cloud and precipitable water content, along with the surface rainfall rate for several rain events in the past. Four cases are considered: (a) TRMM PRs near surface rain rate is correlated with TMI brightness temperatures directly; (b) the total cloud and precipitable water contents along with surface rainfall simulated using WRF are correlated with TMI BTs; (c) similar to case (b) with near surface rainfall taken from TRMM PR measurements; (d) total cloud and precipitable water contents and the surface rain rate are corrected with PR vertical reflectivity profile in a Bayesian framework. The freely available QuickBeam software has been used for simulation of reflectivities at the TRMM PR frequency. The corrected data are then correlated with TMI BTs. Results show that surface rain fall retrievals can be radically improved by using TRMM PR vertical rain corrected total cloud and precipitable water content.

Multiple-Pursuer/One-Evader Pursuit–Evasion Game in Dynamic Flowfields

In this paper, a reachability-based approach is adopted to deal with the pursuit–evasion differential game between one evader and multiple pursuers in the presence of dynamic environmental disturbances (for example, winds or 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 subsequently retrieved from these level sets. The pursuers are divided into active pursuers, guards, and redundant pursuers according to their respective roles in the pursuit–evasion game. The proposed scheme is implemented on problems with both simple and realistic time-dependent flowfields, with and without obstacles.

Energy-optimal path planning in the coastal ocean

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 reanalysis, 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 vehicle-speed 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 the exact particle-backtracking equation. Theoretically, the present methodology is PDE-based and provides fundamental energy-optimal predictions without heuristics. Computationally, it is 3–4 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.

Data-driven learning and modeling of AUV operational characteristics for optimal path planning

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.

Pursuit-evasion games in dynamic flow fields via reachability set analysis

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 readily retrieved 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.

Time-optimal path planning: Real-time sea exercises

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.

A future for intelligent autonomous ocean observing systems

Ocean scientists have dreamed of and recently started to realize an ocean observing revolution with autonomous observing platforms and sensors. Critical questions to be answered by such autonomous systems are where, when, and what to sample for optimal information, and how to optimally reach the sampling locations. Definitions, concepts, and progress towards answering these questions using quantitative predictions and fundamental principles are presented. Results in reachability and path planning, adaptive sampling, machine learning, and teaming machines with scientists are overviewed. The integrated use of differential equations and theory from varied disciplines is emphasized. The results provide an inference engine and knowledge base for expert autonomous observing systems. They are showcased using a set of recent at-sea campaigns and realistic simulations. Real-time experiments with identical autonomous underwater vehicles (AUVs) in the Buzzards Bay and Vineyard Sound region first show that our predicted time-optimal paths were faster than shortest distance paths. Deterministic and probabilistic reachability and path forecasts issued and validated for gliders and floats in the northern Arabian Sea are then presented. Novel Bayesian adaptive sampling for hypothesis testing and optimal learning are finally shown to forecast the observations most informative to estimate the accuracy of model formulations, the values of ecosystem parameters and dynamic fields, and the presence of Lagrangian Coherent Structures.

Autonomy for surface ship interception

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 Marthas 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.