IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. XX, NO. Y, MONTH 2018 1

Graph-Search and Differential Equations forTime-Optimal Vessel Route Planning

in Dynamic Ocean WavesGianandrea Mannarini, Deepak N. Subramani, Pierre F. J. Lermusiaux Member, IEEE, and Nadia Pinardi

Abstract—Time-optimal routes planned by VISIR, a graph-search-based marine vessel route planning system, are evaluatedand compared to the numerical solution of the fundamentaldifferential equations governing time-optimal reachability andpaths in dynamic environments. The comparison exercise em-ploys identical setups: topological constraints, dynamic waveenvironmental conditions, and vessel-ocean parametrizations,while advection by external currents is neglected. The emphasis ison predicting the time-optimal ship headings and Speeds ThroughWater (STW) constrained by dynamic ocean wave fields. SomeVISIR upgrades regarding angular resolution, time-interpolation,and static navigational safety constraints are introduced. For thetime-optimal path planning of a vessel whose STW is constrainedby realistic dynamic waves but whose motion is not advected bystrong external flows, the deviations of the graph-search resultsrelative to the solution of the exact differential equations in boththe path duration and length are found to be of the order of thediscretization errors, and the solutions converge for sufficientresolution.

Index Terms—Optimal vessel heading, graph search method,time-optimal differential optimization, level set equations, reach-ability, ocean modeling, computational performance.

I. INTRODUCTION

Path planning problems are often solved by merging op-timization algorithms with an adequate modeling of the en-vironment and the vehicle’s interaction with it [1]–[3]. Theapproximations and numerical errors of the algorithms usedfor path computation, however, are not often documentednor investigated. This is often due to the exact solution notbeing available, especially for strong and dynamic environ-ments. However, the assessment of such errors is critical forreal-world applications, including optimization of roadwaytravels [4], control of autonomous robots and vehicles inharsh or remote environments such as chemically hazardous

This publication has been developed in cooperation with the EuropeanUnion’s Horizon 2020 research and innovation project AtlantOS (633211)and the Italy-Croatia Interreg V-A programme under the GUTTA project(10043587). The MSEAS team at the Massachusetts Institute of Technology(MIT) thanks the Office of Naval Research (ONR) for research support undergrant N00014-14-1-0476 (Science of Autonomy - LEARNS) and the MIT-TataCenter Program for the Fellowship support of D.N.S..

G. Mannarini is with the Centro Euro-Mediterraneo sui Cambiamenti Cli-matici (CMCC), 73100 Lecce, Italy (e-mail: gianandrea.mannarini@cmcc.it).

D.N. Subramani and P.F.J. Lermusiaux are with the Department of Mechan-ical Engineering, Massachusetts Institute of Technology (MIT), Cambridge,MA 02139 USA (e-mail: deepakns@mit.edu, pierrel@mit.edu).

N. Pinardi is with CMCC and the Universita di Bologna, 40126 Bologna,Italy (e-mail: n.pinardi@sincem.unibo.it).

Manuscript received 2018-08-25; revised: 2019-04-14; accepted .... Date ofpublication ....

environments [2], open and deep ocean [5], extra-terrestrialenvironments [3], climate-optimized aircraft routing [1], andship routing [6]. Comparisons of approximate approaches toexact solutions of optimal path planning would therefore bevery useful. In addition to saving time and energy, trulyoptimal paths reduce the negative effects of transportation onthe environment and its contribution to anthropogenic climatechange [7].

Presently, we focus on the prediction of time-optimal pathsfor marine surface vessels sailing from one location to another,under the influence of a dynamic ocean wave field thatconstraints the surface vessel motions. This is a common issuefor ship operators. Speed loss in waves is due to pitch andheave movements and depends on the vessel hull geometry,cf. App. B. For this effect and also for safety reasons, waveseventually lead to an increase in travel time and energy usageof the vessel.

So far, several approaches have been developed for pathplanning in dynamic environment, with varied level of ap-proximations and accuracy. In one category of approaches,the optimal control problem is formulated on a graph anddynamic programming methods (e.g., [8], [9]), heuristicsearch schemes such as A∗ and Rapidly-exploring RandomTree (RRT, e.g., [10], [11]), nonlinear convex optimization(e.g., [12]–[15]), or evolutionary algorithms (e.g., [16], [17])are employed to solve the optimal control problem. In anothercategory, obstacle avoidance is emphasized and potential fieldmethods [18] or Voronoi diagrams [19] are utilized to identifysafe routes. In yet another category, Fast Marching Methods(FMM, [20], [21]) or wave front expansions ( [22]–[24])are utilized. Recently, the fundamental differential equationsgoverning the reachability front and time- and energy-optimalpaths for vehicles navigating in strong, dynamic, and uncertainenvironments were developed and employed in real settings,e.g., [5], [25]–[29], [68]. Since these differential equationsprovide the exact solution, they are used here to evaluate thesolution provided by a graph-search method.

Graph-search methods are powerful and quite popular invarious R&D sectors (automation [30], text analysis [31], webservices [32], just to mention a few) due to the pervasivepresence of networks in daily life and the fact that theirphysical characteristics have natural mathematical represen-tations, facilitating the understanding even by non-specialists.However, their use for solving time-optimal path or shortestpath problems (SPPs) in strong and dynamic environmentspresents challenges. Starting from [33], it was recognized that

Mannarini, G., D.N. Subramani, P.F.J. Lermusiaux, and N. Pinardi, 2019. Graph-Search and Differential Equations for Time-Optimal Vessel Route Planning in Dynamic Ocean Waves, IEEE Transactions on Intelligent Transportation Systems, sub-judice.