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

Abstract

We develop an accurate partial differential equation based methodology that predicts the timeoptimal 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 doublegyre 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,

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

“…For convenience, we have included a brief 10 description of the methodology, the algorithm, and the 11 relevant notation in §A. A review of the relevant liter-12 ature was previously provided in Lolla et al (2014).…”

Validation of genetic algorithm-based optimal sampling for ocean data assimilation

“…For similar reasons, but also because our planning duration will be limited to a few days, we will not consider the effects of ocean currents in planning the path of vehicles. For such effects of currents, we refer to (Lolla et al, 2014a(Lolla et al, ,b, 2015Subramani et al 2015, Subramani andLermusiaux, 2016) The present genetic-algorithm procedure was originally presented in (Heaney and Duda 2006;Heaney, Gawarkiewicz et al 2007), but was not evaluated within a multiscale ocean environment and in a systematic fashion, using a statistical ensemble of simulations. In this paper, the same approach is applied to an ocean environment that includes multiscale dynamics, from internal tides to mesoscale and larger scale dynamics.…”

“…For general reviews on oceanic path planning, we refer to (Lolla, 2012;Lolla et al 2014a;Lermusiaux et al, 2016) and for general reviews on oceanic adaptive sampling to (Curtin et al 1993;Leonard et al 2007;Lermusiaux, 2007;Roy et al, 2007). Recent efforts for autonomous adaptive sampling include: adaptive sampling via Error Subspace Statistical Estimation (ESSE) with non-linear predictions of error reductions (Lermusiaux 2007); control of coordinated patterns for ocean sampling (Zhang et al, 2007); a mathematical approach to optimally sampling targeted environmental hotspots in the 'MASP uncertainty framework' or multi-robot adaptive sampling problem (Low, et al 2013); Mixed Integer Linear Programming (MILP) for optimal-sampling path planning (Yilmaz et al 2008); nonlinear optimal-sampling path planning using genetic algorithms (Heaney, et al 2007); dynamic programming and onboard routing for optimal-sampling path planning (Wang, et al 2009); command and control of surface kayaks over the Web, directly read from model instructions (Xu et al, 2008); automated sensor networks aiming to facilitate ocean scientific studies (Schofield et al, 2010), and optimal design of glider-sampling networks (Alvarez and Mourre, 2012;Ferri et al, 2015).…”

Time and Energy Optimal Path Planning in General Flows

“…However, for a workspace containing n grid points, the computational cost of the level set method is O(n 3 ) [18] while for graph search method with E connected neighbors, the computational cost is only O(nE log(n)). Furthermore, the level set method periodically performs a re-initialization step, which has a computational cost of O(n 3 ).…”

“…We compared the proposed method with (i) a level set method for planning optimal time paths [18], (ii) a graph search based method for planning optimal energy paths [14].…”