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Computing an Aircraft's Gliding Range and Minimal Return Altitude in Presence of Obstacles and Wind

Giovanni Piccioli

TL;DR

This paper tackles two aviation-planning problems: the Gliding Reachable Region Problem (GRRP) and the Minimal Return Altitude Problem (MRAP), which determine, respectively, the ground region safely reachable in gliding flight and the minimal altitude required to glide to a fixed airfield. By modeling the aircraft with a simplified 3DOF dynamics and incorporating wind and obstacles, the authors derive Hamilton-Jacobi-Bellman equations and solve them on grids using two specialized front-propagation algorithms: Glikonal-G for GRRP and Glikonal-M for MRAP. The methods deliver real-time performance on onboard hardware and provide not only reachability maps but also optimal or feasible trajectories to reach GRR points or airports, respectively. Numerical results on synthetic and real terrain data demonstrate accurate, conservative estimates with robust obstacle avoidance, and highlight practical feasibility for flight-decision support. Limitations include turn-radius effects and wind-turbulence considerations during maneuvers, pointing to future work to integrate turning losses and higher-fidelity dynamics.

Abstract

In the event of a total loss of thrust, a pilot must identify a reachable landing site and subsequently execute a forced landing. To do so, they must estimate which region on the ground can be reached safely in gliding flight. We call this the gliding reachable region (GRR). To compute the GRR, we employ an optimal control formulation aiming to reach a point in space while minimizing altitude loss. A simplified model of the aircraft's dynamics is used, where the effect of turns is neglected. The resulting equations are discretized on a grid and solved numerically. Our algorithm for computing the GRR is fast enough to run in real time during flight, it accounts for ground obstacles and wind, and for each point in the GRR it outputs the path to reach it with minimal loss of altitude. A related problem is estimating the minimal altitude an aircraft needs in order to glide to a given airfield in the presence of obstacles. This information enables pilots to plan routes that always have an airport within gliding distance. We formalize this problem using an optimal control formulation based on the same aircraft dynamics model. The resulting equations are solved with a second algorithm that outputs the minimal re-entry altitude and the paths to reach the airfield from any position while avoiding obstacles. The algorithms we develop are based on the Ordered Upwind Method and the Fast Marching Method.

Computing an Aircraft's Gliding Range and Minimal Return Altitude in Presence of Obstacles and Wind

TL;DR

This paper tackles two aviation-planning problems: the Gliding Reachable Region Problem (GRRP) and the Minimal Return Altitude Problem (MRAP), which determine, respectively, the ground region safely reachable in gliding flight and the minimal altitude required to glide to a fixed airfield. By modeling the aircraft with a simplified 3DOF dynamics and incorporating wind and obstacles, the authors derive Hamilton-Jacobi-Bellman equations and solve them on grids using two specialized front-propagation algorithms: Glikonal-G for GRRP and Glikonal-M for MRAP. The methods deliver real-time performance on onboard hardware and provide not only reachability maps but also optimal or feasible trajectories to reach GRR points or airports, respectively. Numerical results on synthetic and real terrain data demonstrate accurate, conservative estimates with robust obstacle avoidance, and highlight practical feasibility for flight-decision support. Limitations include turn-radius effects and wind-turbulence considerations during maneuvers, pointing to future work to integrate turning losses and higher-fidelity dynamics.

Abstract

In the event of a total loss of thrust, a pilot must identify a reachable landing site and subsequently execute a forced landing. To do so, they must estimate which region on the ground can be reached safely in gliding flight. We call this the gliding reachable region (GRR). To compute the GRR, we employ an optimal control formulation aiming to reach a point in space while minimizing altitude loss. A simplified model of the aircraft's dynamics is used, where the effect of turns is neglected. The resulting equations are discretized on a grid and solved numerically. Our algorithm for computing the GRR is fast enough to run in real time during flight, it accounts for ground obstacles and wind, and for each point in the GRR it outputs the path to reach it with minimal loss of altitude. A related problem is estimating the minimal altitude an aircraft needs in order to glide to a given airfield in the presence of obstacles. This information enables pilots to plan routes that always have an airport within gliding distance. We formalize this problem using an optimal control formulation based on the same aircraft dynamics model. The resulting equations are solved with a second algorithm that outputs the minimal re-entry altitude and the paths to reach the airfield from any position while avoiding obstacles. The algorithms we develop are based on the Ordered Upwind Method and the Fast Marching Method.
Paper Structure (34 sections, 2 theorems, 41 equations, 18 figures, 3 algorithms)

This paper contains 34 sections, 2 theorems, 41 equations, 18 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Our contribution
  4. Related literature
  5. Problem definition
  6. Aircraft dynamics
  7. Gliding Reachable Region Problem
  8. Optimal paths
  9. Gliding as front propagation
  10. Minimal return altitude problem
  11. Algorithms
  12. Numerical results
  13. Glikonal-G
  14. GRRP on real data
  15. Computational efficiency
  16. ...and 19 more sections

Key Result

Lemma A.1

For all $\bm y\in\mathop{\mathrm{\mathbb{R}}}\nolimits^2$, $V_2(\bm y)=V(\bm y)$.

Figures (18)

  • Figure 1: Result of running Glikonal-G on flat terrain with uniform wind. The red dot represents the initial position of the glider. (a) Heatmap and contour lines of the function $z_0-U_G$. The red region is outside the gliding reachable region. The turquoise lines are the optimal trajectories. (b) Relative error of the Glikonal-G solution, i.e., $(U_G-U)/U$. (c) 3d plot of the function $z_0-U_G$.
  • Figure 2: Glikonal-G's solution in the case of non-uniform wind and single mountain peak. The red dot represents the glider's initial position. (a) Wind direction (constant with altitude) plotted on top of the elevation profile (displayed as a heatmap). (b) Contour lines of the function $z_0-U_G$ (in white) and optimal trajectories (in turquoise). The heatmap is the elevation profile, and the red-shaded regions are those not reachable in gliding flight. (c) 3d representation of the function $z_0-U_G$, in orange and of $h_\text{min}$.
  • Figure 3: Glikonal-G on a real-world example. Horizontal distance is measured in kilometers, while the altitude is in meters (see colorbar). The red and purple dots represent respectively the glider's and airfield's positions. (a) Heatmap of the elevation profile of the Jura mountain range. The glider and the closest airfield are located on opposite sides of the range. The white arrows represent the uniform wind across the map. (b) Contour lines of the altitude function $z_0-U_G$ in black and optimal trajectories in blue. The red-shaded region is unreachable. Background map from OpenStreetMap. (c) 3d representation of the altitude of the glider ($z_0-U_G$) in orange, on top of the elevation profile. for visualization purposes, the vertical axis is rescaled by 8 in comparison to the horizontal axes. The purple vertical line highlights the position of the airfield.
  • Figure 4: Numerical experiments on Glikonal-M. The purple dot indicates the position of the airfield. Top row: results on flat terrain. (a) Heatmap and contour lines of the function $V_G$. The turquoise lines are feasible re-entry trajectories to the airfield. (b) Relative error of the Glikonal-M solution, i.e. $(V_G-V)/V$. (c) 3d plot of the function $V_G$. Bottom row: elevation profile with a single mountain peak. (d) Heatmap of the elevation profile with contour lines of the function $V_G$ (in white) and re-entry trajectories (in turquoise). (e) 3d plot of the function $V_G$ in orange on top of the elevation profile.
  • Figure 5: Glikonal-M on a real-world example. The horizontal distances are measured in kilometers. The purple dot represents the airfield's position. (a) Contour lines of the minimal altitude function $V_G$ in black and feasible trajectories in blue. (b) 3d representation of the function $V_G$ in orange, on top of the elevation profile. For visualization purposes, the vertical axis is rescaled by 8 in comparison to the horizontal axes. The purple vertical line highlights the position of the airfield.
  • ...and 13 more figures

Theorems & Definitions (4)

  • Lemma A.1
  • proof
  • Lemma A.2
  • proof