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Study of a numerical optimization of a flight trajectory

Author: Daniel Colás Irazusta

Director: Alex Ferrer Ferre

Degree: Bachelor in Aerospace Vehicle Engineering

Examination session: Spring 2024

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Table of contents

  1. Introduction
  2. Mathematical formulation
  3. Launch of an object without drag
  4. Launch of an object considering drag
  5. Flight equations
    1. Case 1: Airbus 320 climb
    2. Case 2: Airbus 320 descend
  6. Conclusions and future lines of research

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Introduction

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[1] Hannah Ritchie (2020) - “Sector by sector: where do global greenhouse gas emissions come from?” Published online at OurWorldInData.org. Retrieved from: 'https://ourworldindata.org/ghg-emissions-by-sector' [Online Resource]

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Mathematical formulation

  • Use of the Lagrange multipliers

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Mathematical formulation

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  • Langrangian
  • State conditions

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Mathematical formulation

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  • Differentiation of the Langrangian

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  • Adjoint problem

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Mathematical formulation

  • Differentiation of the Langrangian

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  • Update of the input parameters

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Mathematical formulation

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Solving the ODE system

Calculation of (Adjoint problem)

Optimized result

If tolerance is not meet

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Launch of an object without drag

  • Objective:
    • Maximize distance
  • Variables:

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  • Constrains:

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Launch of an object without drag

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Simulation

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Iterations

1

15

18.77

454

2

15

40.508

296

3

15

74.782

245

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Launch of an object considering drag

  • Objective:
    • Maximize distance
  • Variables:

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  • Constrains:

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Launch of an object considering drag

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Simulation

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Iterations

1

15

19.887

331

2

15

48.868

210

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Launch of an object considering drag

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Launch of an object considering drag

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Flight equations

  • Objective:
    • Maximize final weight
  • Variables:

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  • Constrains:

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Flight equations

  • Air density
    • Below 11,000 m

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  • Mach number

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  • Above 11,000 m

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Flight equations

  • Lift

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  • Drag

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  • Thrust
  • Lift coefficient

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  • Drag coefficient

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  • Thrust at see level

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Case 1: Airbus 320 climb

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Case 1: Airbus 320 climb

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Case 1: Airbus 320 climb

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Case 2: Airbus 320 descend

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Case 2: Airbus 320 descend

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Case 2: Airbus 320 descend

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Conclusions

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  • Development an numerical optimization algorithm

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  • Resolution of the optimization of the launch of an object

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  • Resolution of the optimization of a climb and a descend

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Future lines of research

  • Application in different cases
    • Optimization of the rocked equations
    • Optimization of orbital mechanics maneuvers

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  • Expansion to a three dimensions problem
    • Following departure and arrival maneuvers of airports
    • Following most effective route given two points

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[2] “AIP España.” [Online]. Published online at enaire.es. Retrieved from: https://aip.enaire.es/AIP/contenido_AIP/AD/AD2/LEBL/LE_AD_2_LEBL_SID_5_en.pdf