Tapovan Lolla

Path planning in time dependent flow fields using level set methods

We develop and illustrate an efficient but rigorous methodology that predicts the time-optimal paths of ocean vehicles in continuous dynamic flows. The goal is to best utilize or avoid currents, without limitation on these currents or on the number of vehicles. The methodology employs a new modified level set equation to evolve a front from the starting point of a vehicle until it reaches the desired goal location, combining flow advection with nominal vehicle motion. The optimal path of the vehicle is then obtained by solving a particle tracking equation backward in time. The computational cost of this method increases linearly with the number of vehicles and geometrically with spatial dimensions. The methodology is applicable to any continuous flow and in scenarios with multiple vehicles. Present illustrations consist of the crossing of a canonical uniform jet and its validation using a classic optimization solution, as well as swarm formation in more complex time varying 2D flow fields, including jets, eddies and forbidden regions.

Time-optimal path planning in dynamic flows using level set equations: theory and schemes

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.

Time-optimal path planning in dynamic flows using level set equations: realistic applications

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.

Science of Autonomy: Time-Optimal Path Planning and Adaptive Sampling for Swarms of Ocean Vehicles

The science of autonomy is the systematic development of fundamental knowledge about autonomous decision making and task completing in the form of testable autonomous methods, models and systems. In ocean applications, it involves varied disciplines that are not often connected. However, marine autonomy applications are rapidly growing, both in numbers and in complexity. This new paradigm in ocean science and operations motivates the need to carry out interdisciplinary research in the science of autonomy. This chapter reviews some recent results and research directions in time-optimal path planning and optimal adaptive sampling. The aim is to set a basis for a large number of vehicles forming heterogeneous and collaborative underwater swarms that are smart, i. e., knowledgeable about the predicted environment and their uncertainties, and about the predicted effects of autonomous sensing on future operations. The methodologies are generic and applicable to any swarm that moves and senses dynamic environmental fields. However, our focus is underwater path planning and adaptive sampling with a range of vehicles such as autonomous underwater vehicles (AUV s), gliders, ships or remote sensing platforms.

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.

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.

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.

A Gaussian Mixture Model Smoother for Continuous Nonlinear Stochastic Dynamical Systems: Theory and Scheme

AbstractRetrospective inference through Bayesian smoothing is indispensable in geophysics, with crucial applications in ocean and numerical weather estimation, climate dynamics, and Earth system modeling. However, dealing with the high-dimensionality and nonlinearity of geophysical processes remains a major challenge in the development of Bayesian smoothers. Addressing this issue, a novel subspace smoothing methodology for high-dimensional stochastic fields governed by general nonlinear dynamics is obtained. Building on recent Bayesian filters and classic Kalman smoothers, the fundamental equations and forward–backward algorithms of new Gaussian Mixture Model (GMM) smoothers are derived, for both the full state space and dynamic subspace. For the latter, the stochastic Dynamically Orthogonal (DO) field equations and their time-evolving stochastic subspace are employed to predict the prior subspace probabilities. Bayesian inference, both forward and backward in time, is then analytically carried out in the...

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.