Energy-optimal path planning by stochastic dynamically orthogonal level-set optimization

Abstract

A stochastic optimization methodology is formulated for computing energy-optimal paths from among time-optimal paths of autonomous vehicles navigating in a dynamic flow field. Based on partial differential equations, the methodology rigorously leverages the level-set equation that governs time-optimal reachability fronts for a given relative vehicle-speed function. To set up the energy optimization, the relative vehicle-speed and headings are considered to be stochastic and new stochastic Dynamically Orthogonal (DO) level-set equations are derived. Their solution provides the distribution of time-optimal reachability fronts and corresponding distribution of time-optimal paths. An optimization is then performed on the vehicle's energy-time joint distribution to select the energy-optimal paths for each arrival time, among all stochastic time-optimal paths for that arrival time. Numerical schemes to solve the reduced stochastic DO level-set equations are obtained, and accuracy and efficiency considerations are discussed. These reduced equations are first shown to be efficient at solving the governing stochastic level-sets, in part by comparisons with direct Monte Carlo simulations. To validate the methodology and illustrate its accuracy, comparisons with 'semi-analytical' energy-optimal path solutions are then completed. In particular, we consider the energy-optimal crossing of a canonical steady front and set up its 'semi-analytical' solution using a energy-time nested nonlinear double-optimization scheme. We then showcase the inner workings and nuances of the energy-optimal path planning, considering different mission scenarios. Finally, we study and discuss results of energy-optimal missions in a wind-driven barotropic quasi-geostrophic double-gyre ocean circulation.

Energy‐optimal path planning in the coastal ocean

“…Only the zero level‐set is needed, but for numerical convenience, an open boundary condition can be used at the numerical domain boundaries , e.g., , where n is the outward normal to . The subsequent solution to the backtracking (equation ), yields the continuous‐time history of the time‐optimal vehicle heading angles, [ Lolla et al ., ; Lolla and Lermusiaux , ; Subramani and Lermusiaux , ].…”

Optimal Planning and Sampling Predictions for Autonomous and Lagrangian Platforms and Sensors in the Northern Arabian Sea

“…They were theoretically extended to anisotropic motions and to fully three-dimensional paths, including planning for floats or other vehicles with constrained relative motions. They were also used successfully with real AUVs at sea (Subramani et al, in press; Edwards et al, in press) and extended to uncertain stochastic currents (Wei, 2015) and to energy-optimal path planning (Subramani and Lermusiaux, 2016;.…”

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

“…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).…”

“…This is not done here, as it involves an additional level of complexity (e.g. Lermusiaux et al, 2016;Lolla, 2016) to an adaptive sampling cost function that is already complex. 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 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.…”