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An Eikonal Approach for Globally Optimal Free Flight Trajectories

Ralf Borndörfer, Arturas Jocas, Martin Weiser

Abstract

We present an eikonal-based approach that is capable of finding a continuous globally optimal trajectory for an aircraft in a stationary wind field. This minimizes emissions and fuel consumption. If the destination is close to a cut locus of the associated Hamilton-Jacobi-Bellman equation, small numerical discretization errors can lead to selecting a merely locally optimal trajectory and missing the globally optimal one. Based on finite element error estimates, we construct a trust region around the cut loci in order to guarantee uniqueness of trajectories for destinations sufficiently far from cut loci.

An Eikonal Approach for Globally Optimal Free Flight Trajectories

Abstract

We present an eikonal-based approach that is capable of finding a continuous globally optimal trajectory for an aircraft in a stationary wind field. This minimizes emissions and fuel consumption. If the destination is close to a cut locus of the associated Hamilton-Jacobi-Bellman equation, small numerical discretization errors can lead to selecting a merely locally optimal trajectory and missing the globally optimal one. Based on finite element error estimates, we construct a trust region around the cut loci in order to guarantee uniqueness of trajectories for destinations sufficiently far from cut loci.
Paper Structure (8 sections, 5 theorems, 28 equations, 3 figures)

This paper contains 8 sections, 5 theorems, 28 equations, 3 figures.

Table of Contents

  1. Introduction
  2. An Eikonal Approach
  3. Regularity of Hamilton-Jacobi-Bellman Solutions
  4. Discretization
  5. Localizing the Singularities
  6. Numerical Results
  7. Acknowledgments
  8. Appendix

Key Result

theorem 1

PIGNOTTI2002681 Let $u$ be a solution of chapter2:hjb_pde. Denote the set of regular points as $\mathcal{R} = \mathbb{R}^2 \setminus \{ \mathcal{K} \cup \Sigma \cup \Gamma \}$. If $w \in C^3(\mathbb{R}^2, \mathbb{R}^2)$ and $\partial \mathcal{K}$ is $C^3$, then $\Sigma \cup \Gamma$ is closed and the

Figures (3)

  • Figure 1: Optimal trajectories from the blue origin to two different red destination points in a single-vortex wind field are indicated as solid black lines; A, C, D are globally optimal, B is only locally optimal. Estimated singularities, so called cut loci, which can be reached by multiple globally optimal trajectories, are shown as a red line. The blue region indicates a (scaled) trust region around the cut loci; the magenta region depicts the region around the set of conjugate points, see Def. \ref{['def:nearly_conj_pt']} below
  • Figure 2: The discretization errors are displayed against their estimates for cases (a)--(d) with varying FE mesh size
  • Figure 3: Visualization of the temporal $2 \varepsilon$ trust region (blue points) around $\Sigma_h \cup \Gamma_h$ (blue triangles) computed using an FE solution with a $201 \times 201$ resolution. The FE solution with a $1001 \times 1001$ resolution is used as a reference solution to compute the a posteriori discretization error $\varepsilon$ and the cut loci $\Sigma \cup \Gamma$ (red triangles). The right plot is a zoomed-in version of the left plot depicting a positional shift in $\Sigma_h \cup \Gamma_h$

Theorems & Definitions (12)

  • definition 1
  • theorem 1
  • definition 2: $\tau$-nearly Conjugate Points
  • theorem 2: Regularity
  • proof
  • theorem 3: Discretization Error Rasch2007
  • proof
  • theorem 4: Cut Loci Position
  • proof
  • Remark 1
  • ...and 2 more