Numerical Techniques for Minimum-Time Routing on Sphere with Realistic Winds

Marchidan, Andrei; Bakolas, Efstathios

doi:10.2514/1.g001389

Cited by 15 publications

(6 citation statements)

References 21 publications

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“…In the field of optimal control, the Zermelo problem is usually taken with application of Pontryagin's maximum principle (Pontryagin et al, 1962). In what follows we proceed via the Lagrangian and this way is in general equivalent to the discussion on the navigation problem via the Hamiltonian; see Bijlsma (2001, 2009), Hull (2009), Techy and Woolsey (2009), Bijlsma (2010), Techy (2011), Jardin and Bryson (2012), Burns (2013), Li et al (2013) and Marchidan and Bakolas (2016). However, which of them is more convenient in the sense of solution and computational complexity depends in fact on the specific navigation data.…”

Section: Preliminariesmentioning

confidence: 99%

On Generalised Single-Heading Navigation

Aldea

Kopacz

2020

J. Navigation

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Introducing the notion of a pseudoloxodrome, we generalise a single-heading navigation to conformally flat Riemannian manifolds, under the action of a perturbing vector field (wind, current) of arbitrary force. The findings are applied to time-optimal navigation with the use of the Euler–Lagrange equations. We refer to the Zermelo navigation problem admitting space and time dependence of both a perturbation and a ship's speed. The necessary conditions for single-heading time-optimal navigation are obtained and the pseudoloxodromes of minimum and maximum time are discussed. Furthermore, we describe winds which yield the pseudoloxodromic and loxodromic time extremals. Our research is also illustrated with the examples in dimension two emphasising the single-heading solutions among the time-optimal trajectories in the presence of some space-dependent winds.

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Section: Preliminariesmentioning

confidence: 99%

On Generalised Single-Heading Navigation

Aldea

Kopacz

2020

J. Navigation

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“…Calculation of travel time is a focal research study in the domain of air traffic control. By relaxing kinodynamic compatibility and obstacle avoidance constraints, research studies aim to find the optimal horizontal trajectory (flying in a constant altitude) taking into account wind (see, Bryson, 2001, 2012;Marchidan and Bakolas, 2015;Girardet et al, 2014;Marchidan and Bakolas, 2015;Bonami et al, 2013).…”

Section: Travel Time Calculationmentioning

confidence: 99%

A two-stage route optimization algorithm for light aircraft transport systems

Azadeh

Bierlaire

Maknoon

2019

Transportation Research Part C: Emerging Technologies

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“…It, however, does not deal with optimality of the path in terms of length. In [4], a numerical technique is presented to generate optimal trajectories on a sphere in the presence of wind, however, this does not consider the geodesic curvature constraints. When the motion is constrained to the surface of a sphere, a natural generalization can be provided through the geodesic curvature constraint on the motion, where the geodesic curvature constraint is the counterpart of the planar curvature constraint on the motion of Dubins vehicle.…”

Section: Introductionmentioning

confidence: 99%

Optimal Geodesic Curvature Constrained Dubins' Paths on a Sphere

Darbha¹,

Athindra²,

Rajagopal³

et al. 2022

Preprint

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In this article, we consider the motion planning of a rigid object on the unit sphere with a unit speed. The motion of the object is constrained by the maximum absolute value, Umax of geodesic curvature of its path; this constrains the object to change the heading at the fastest rate only when traveling on a tight smaller circular arc of radius r < 1, where r depends on the bound, Umax. We show in this article that if 0 < r ≤ 1 2 , the shortest path between any two configurations of the rigid body on the sphere consists of a concatenation of at most three circular arcs. Specifically, if C is the smaller circular arc and G is the great circular arc, then the optimal path can only be CCC, CGC, CC, CG, GC, C or G. If r > 1 2 , while paths of the above type may cease to exist depending on the boundary conditions and the value of r, optimal paths may be concatenations of more than three circular arcs. Mathematical Formulation ModelConsider the trajectory of a rigid massless robot (modeled as a short pointed stick) on a unit sphere as shown by the green spherical curve in Fig. 1.Let s denote arc length (traversed by the robot), and X(s) be a C 1 -smooth spherical curve, representing the position of the robot (geometric center of the stick) on the sphere. Let T(s) := dX ds represent the direction

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Numerical Techniques for Minimum-Time Routing on Sphere with Realistic Winds

Cited by 15 publications

References 21 publications

On Generalised Single-Heading Navigation

On Generalised Single-Heading Navigation

A two-stage route optimization algorithm for light aircraft transport systems

Optimal Geodesic Curvature Constrained Dubins' Paths on a Sphere

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