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A Discrete-Continuous Algorithm for Globally Optimal Free Flight Trajectory Optimization

Authors Ralf Borndörfer , Fabian Danecker , Martin Weiser



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Author Details

Ralf Borndörfer
  • Zuse Institute, Berlin, Germany
  • Freie Universität Berlin, Germany
Fabian Danecker
  • Zuse Institute, Berlin, Germany
Martin Weiser
  • Zuse Institute, Berlin, Germany

Acknowledgements

We thank three anonymous referees for helpful comments that improved the presentation of this paper.

Cite AsGet BibTex

Ralf Borndörfer, Fabian Danecker, and Martin Weiser. A Discrete-Continuous Algorithm for Globally Optimal Free Flight Trajectory Optimization. In 22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022). Open Access Series in Informatics (OASIcs), Volume 106, pp. 2:1-2:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/OASIcs.ATMOS.2022.2

Abstract

We present an efficient algorithm that finds a globally optimal solution to the 2D Free Flight Trajectory Optimization Problem (aka Zermelo Navigation Problem) up to arbitrary precision in finite time. The algorithm combines a discrete and a continuous optimization phase. In the discrete phase, a set of candidate paths that densely covers the trajectory space is created on a directed auxiliary graph. Then Yen’s algorithm provides a promising set of discrete candidate paths which subsequently undergo a locally convergent refinement stage. Provided that the auxiliary graph is sufficiently dense, the method finds a path that lies within the convex domain around the global minimizer. From this starting point, the second stage will converge rapidly to the optimum. The density of the auxiliary graph depends solely on the wind field, and not on the accuracy of the solution, such that the method inherits the superior asymptotic convergence properties of the optimal control stage.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Continuous functions
  • Mathematics of computing → Discretization
  • Mathematics of computing → Discrete optimization
  • Mathematics of computing → Continuous optimization
  • Mathematics of computing → Nonconvex optimization
  • Mathematics of computing → Graph algorithms
Keywords
  • shortest path
  • flight planning
  • free flight
  • discretization error bounds
  • optimal control
  • discrete optimization
  • global optimization

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References

  1. Bernardetta Addis, Andrea Cassioli, Marco Locatelli, and Fabio Schoen. A global optimization method for the design of space trajectories. Computational Optimization and Applications, 48:635-652, 2011. URL: https://doi.org/10.1007/s10589-009-9261-6.
  2. Ali Al Zoobi, David Coudert, and Nicolas Nisse. On the complexity of finding k shortest dissimilar paths in a graph. Research report, Inria ; CNRS ; I3S ; Université Côte d'Azur, 2021. URL: https://hal.archives-ouvertes.fr/hal-03187276.
  3. Ioannis P. Androulakis, Costas D. Maranas, and Christodoulos A. Floudas. αBB: A global optimization method for general constrained nonconvex problems. Journal of Global Optimization, 7(4):337-363, 1995. URL: https://doi.org/10.1007/BF01099647.
  4. Marco Blanco, Ralf Borndörfer, Nam-Dung Hoang, Anton Kaier, Pedro Maristany de las Casas, Thomas Schlechte, and Swen Schlobach. Cost Projection Methods for the Shortest Path Problem with Crossing Costs. In Atmos, 2017. URL: https://doi.org/10.4230/OASIcs.ATMOS.2017.15.
  5. Marco Blanco, Ralf Borndörfer, Nam-Dung Hoang, Anton Kaier, Adam Schienle, Thomas Schlechte, and Swen Schlobach. Solving Time Dependent Shortest Path Problems on Airway Networks Using Super-Optimal Wind. In 16th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2016). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. URL: https://doi.org/10.4230/OASIcs.ATMOS.2016.12.
  6. Mohammad Reza Bonyadi and Zbigniew Michalewicz. Particle Swarm Optimization for Single Objective Continuous Space Problems: A Review. Evolutionary Computation, 25(1):1-54, 2017. URL: https://doi.org/10.1162/EVCO_r_00180.
  7. Ralf Borndörfer, Fabian Danecker, and Martin Weiser. A Discrete-Continuous Algorithm for Free Flight Planning. Algorithms, 14(1):4, 2021. URL: https://doi.org/10.3390/a14010004.
  8. Ralf Borndörfer, Fabian Danecker, and Martin Weiser. Error bounds for Discrete-Continuous Shortest Path Problems with Application to Free Flight Trajectory Optimization. under revision, 2021. Google Scholar
  9. Dietrich Braess. Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie. Springer-Verlag, 2013. URL: https://doi.org/10.1007/978-3-642-34797-9.
  10. Andrea Cassioli, Dario Izzo, David Lorenzo, Marco Locatelli, and Fabio Schoen. Global Optimization Approaches for Optimal Trajectory Planning, pages 111-140. Springer, 2012. URL: https://doi.org/10.1007/978-1-4614-4469-5_5.
  11. Huite M. de Jong. Optimal Track Selection and 3-dimensional flight planning. techreport, Royal Netherlands Meteorological Institute, 1974. Google Scholar
  12. Holger Diedam. Global optimal control using direct multiple shooting. PhD thesis, Heidelberg University, 2015. Google Scholar
  13. Gang Feng. Finding k shortest simple paths in directed graphs: A node classification algorithm. Networks, 64(1):6-17, 2014. URL: https://doi.org/10.1002/net.21552.
  14. Christodoulos A Floudas. Deterministic global optimization: theory, methods and applications, volume 37. Springer Science & Business Media, 2013. Google Scholar
  15. Bernd Hartke. Global optimization. Wiley Interdisciplinary Reviews: Computational Molecular Science, 1(6):879-887, 2011. URL: https://doi.org/10.1002/wcms.70.
  16. Casper Kehlet Jensen, Marco Chiarandini, and Kim S. Larsen. Flight Planning in Free Route Airspaces. In Gianlorenzo D'Angelo and Twan Dollevoet, editors, 17th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2017), volume 59 of OpenAccess Series in Informatics (OASIcs), pages 1-14, 2017. URL: https://doi.org/10.4230/OASIcs.ATMOS.2017.14.
  17. Donald R. Jones, Cary D. Perttunen, and Biruce E. Stuckman. Lipschitzian Optimization without the Lipschitz Constant. J. Optim. Theory Appl., 79(1):157-181, 1993. URL: https://doi.org/10.1007/BF00941892.
  18. Sertac Karaman and Emilio Frazzoli. Sampling-based algorithms for optimal motion planning. The international journal of robotics research, 30(7):846-894, 2011. URL: https://doi.org/10.1177/0278364911406761.
  19. Stefan E. Karisch, Stephen S. Altus, Goran Stojković, and Mirela Stojković. Operations. In Quantitative problem solving methods in the airline industry, pages 283-383. Springer, 2012. URL: https://doi.org/10.1007/978-1-4614-1608-1_6.
  20. Mohammad Khajehzadeh, Mohd Raihan Taha, Ahmed El-Shafie, and Mahdiyeh Eslami. A survey on meta-heuristic global optimization algorithms. Research Journal of Applied Sciences, Engineering and Technology, 3(6):569-578, 2011. Google Scholar
  21. Anders N. Knudsen, Marco Chiarandini, and Kim S. Larsen. Heuristic Variants of A^*-Search for 3D Flight Planning. In Willem-Jan van Hoeve, editor, Integration of Constraint Programming, Artificial Intelligence, and Operations Research, pages 361-376, Cham, 2018. Springer International Publishing. URL: https://doi.org/10.1007/978-3-319-93031-2_26.
  22. Anders Nicolai Knudsen, Marco Chiarandini, and Kim S. Larsen. Constraint Handling in Flight Planning. In Principles and Practice of Constraint Programming - 23superscriptrd International Conference, CP 2017, Melbourne, VIC, Australia, August 28 - September 1, 2017, Proceedings, pages 354-369, 2017. URL: https://doi.org/10.1007/978-3-319-66158-2_23.
  23. Harold W. Kuhn and Albert W. Tucker. Nonlinear programming. In Proceedings of the second Berkeley Symposium on Mathematical Statistics and Probability, pages 481-493. University of California Press, 1951. Google Scholar
  24. Denis Kurz and Petra Mutzel. A sidetrack-based algorithm for finding the k shortest simple paths in a directed graph. In Seok-Hee Hong, editor, 27th International Symposium on Algorithms and Computation, ISAAC 2016, December 12-14, 2016, Sydney, Australia, volume 64 of LIPIcs, pages 49:1-49:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2016.49.
  25. Marco Locatelli. Simulated Annealing Algorithms for Continuous Global Optimization. In Handbook of global optimization, pages 179-229. Springer, 2002. URL: https://doi.org/10.1007/978-1-4757-5362-2_6.
  26. Thi Thoa Mac, Cosmin Copot, Duc Trung Tran, and Robin De Keyser. Heuristic approaches in robot path planning: A survey. Robotics and Autonomous Systems, 86:13-28, 2016. URL: https://doi.org/10.1016/j.robot.2016.08.001.
  27. Amgad Madkour, Walid G. Aref, Faizan Ur Rehman, Mohamed Abdur Rahman, and Saleh Basalamah. A Survey of Shortest-Path Algorithms, 2017. URL: http://arxiv.org/abs/1705.02044.
  28. Hok K. Ng, Banavar Sridhar, and Shon Grabbe. Optimizing Aircraft Trajectories with Multiple Cruise Altitudes in the Presence of Winds. Journal of Aerospace Information Systems, 11(1):35-47, 2014. URL: https://doi.org/10.2514/1.I010084.
  29. Adam Schienle, Pedro Maristany de las Casas, and Marco Blanco. A Priori Search Space Pruning in the Flight Planning Problem. In 19th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2019), volume 75 of OpenAccess Series in Informatics (OASIcs), pages 8:1-8:14, 2019. URL: https://doi.org/10.4230/OASIcs.ATMOS.2019.8.
  30. Omar Souissi, Rabie Benatitallah, David Duvivier, AbedlHakim Artiba, Nicolas Belanger, and Pierre Feyzeau. Path planning: A 2013 survey. In Proceedings of 2013 International Conference on Industrial Engineering and Systems Management (IESM), pages 1-8. IEEE, 2013. Google Scholar
  31. Cathie A. Wells, Paul D. Williams, Nancy K. Nichols, Dante Kalise, and Ian Poll. Reducing transatlantic flight emissions by fuel-optimised routing. Environmental Research Letters, 16(2):025002, 2021. URL: https://doi.org/10.1088/1748-9326/abce82.
  32. Jin Y. Yen. Finding the K Shortest Loopless Paths in a Network. Management Science, 17(11):712-716, 1971. URL: https://doi.org/10.1287/mnsc.17.11.712.
  33. Ernst Zermelo. Über das Navigationsproblem bei ruhender oder veränderlicher Windverteilung. ZAMM Z. Angew. Math. Mech., 11(2):114-124, 1931. URL: https://doi.org/10.1002/zamm.19310110205.
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