Time-dependent Zermelo navigation with tacking

TL;DR

This paper extends Zermelo navigation theory to time-dependent and non-convex velocity profiles, revealing new properties, tacking behaviors, and providing efficient algorithms for computing optimal trajectories in complex environments.

Contribution

It introduces novel global results for time-dependent indicatrix fields and explores non-convex cases with tacking, offering new insights and computational methods for Zermelo navigation problems.

Findings

Time-dependent indicatrix fields exhibit favorable properties for navigation.

Non-convex indicatrices can lead to zig-zag tacking trajectories.

New algorithms effectively compute time-minimizing paths in complex scenarios.

Abstract

We address the time- and position-dependent Zermelo navigation problem within the framework of Lorentz-Finsler geometry. Since the initial work of E. Zermelo, the task is to find the time-minimizing trajectory between two regions for a moving object whose speed profile depends on time, position and direction. We give a step-by-step review of the classical formulation of the problem, where the geometric shape generated by the velocity vectors -- the speed profile indicatrix -- is strongly convex at each point. We derive new global results for the cases where the indicatrix field is only time-dependent. In such (meso-scale realistic) cases, Zermelo navigation exhibits particularly favorable properties that have not been previously explored, making them especially appealing for both theoretical and numerical investigations. Moreover, motivated by real-world phenomena and examples, we obtain novel results for non-convex (or multi-convex) navigation, i.e. when the indicatrices fail to be convex. In this new setting -- which is not unlike the corresponding Finsler setting for Snell's law -- optimal paths may involve so-called tacking, which stems from discontinuous shifts of direction of motion. The tacking behaviour thus results in zig-zag trajectories, as observed in time-optimal sailboat navigation and surprisingly also in the flight paths of far-ranging seabirds. Finally, we provide new efficient computational algorithms and illustrate the use of them to solve the Zermelo navigation (boundary value) problem, i.e. to find numerically the time-minimizing trajectory between two fixed points in the general non-convex setting.