Abstract
Air traffic management ensures the safety of flight by optimizing flows and maintaining separation between aircraft. After giving some definitions, some typical feature of aircraft trajectories are presented. Trajectories are objects belonging to spaces with infinite dimensions. The naive way to address such problem is to sample trajectories at some regular points and to create a big vector of positions (and or speeds). In order to manipulate such objects with algorithms, one must reduce the dimension of the search space by using more efficient representations. Some dimension reduction tricks are then presented for which advantages and drawbacks are presented. Then, front propagation approaches are introduced with a focus on Fast Marching Algorithms and Ordered upwind algorithms. An example of application of such algorithm to a real instance of air traffic control problem is also given. When aircraft dynamics have to be included in the model, optimal control approaches are really efficient. We present also some application to aircraft trajectory design. Finally, we introduce some path planning techniques via natural language processing and mathematical programming.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
TMA: “Terminal Maneuvering Area”.
- 2.
The convex hull or convex envelope of a set X of points in the Euclidean plane or Euclidean space is the smallest convex set that contains X.
References
Atkins EM, Portillo IA, Strube MJ (2006) Emergency flight planning applied to total loss of thrust. J Guid Control Dyn 43(4):1205–1216
Bakolas E, Zhao Y, Tsiotras P (2011) Initial guess generation for aircraft landing trajectory optimization. In: AIAA guidance, navigation, and control conference. AIAA
Bartels RH, Beatty JC, Barskyn BA (1998) An introduction to splines for use in computer graphics and geometric modeling. Computer graphics. Morgan Kaufmann, San Francisco
Becerra VM (2011) Psopt optimal control solver user manual. Tech. report
Bellingham J, Kuwata Y, How J (2003) Stable receding horizon trajectory control for complex environment. In: AIAA guidance, navigation, and control conference and exhibit. AIAA
Berger M, Gostiaux B (1988) Differential geometry: manifolds, curves and surfaces. Springer, New York
Betts JT (1998) Survey of numerical methods for trajectory optimization. J Guid Control Dyn 21(2):193–207
Betts JT, Huffman WP (1997) Sparse optimal control software SOCS. Tech. report, Mathematics and Engineering Analysis Technical Document MEALR-085, Boeing Information and Support Services, The Boeing Company
Betts JT, Huffman WP (1998) Mesh refinement in direct transcription methods for optimal control. Optim Control Appl Methods 19(1):1–21
Betts JT, Biehn N, Campbell SL, Huffman WP (2000) Compensating for order variation in mesh refinement for direct transcription methods. J Comput Appl Math 125:147–158
Binder T, Blank L, Dahmen W, Marquardt W (2000) Grid refinement in multiscale dynamic optimization. Tech. report, RWTH Aachen
Birkhoff G, de Boor C (1964) Piecewise polynomial interpolation and approximation. In: Proceeding of the general motors symposium of 1964. General Motors
Brudnicki DJ, McFarland AL (1997) User request evaluation tool (uret) conflict probe performance and benefits assessment. In: Proceeding of the air traffic management seminar, FAA/Eurocontrol
Bulirsch R, Montrone F, Pesch HJ (1991) Abort landing in the presence of windshear as a minimax optimal control problem. Part I: necessary conditions. J Optim Theory Appl 70(1):1–23
Bulirsch R, Montrone F, Pesch HJ (1991) Abort landing in the presence of windshear as a minimax optimal control problem. Part II: multiple shooting and homotopy. J Optim Theory Appl 70(2):223–254
Burrows JW (1983) Fuel-optimal aircraft trajectories with fixed arrival times. J Guid Control Dyn 6(1):14–19
Chaimatanan S, Delahaye D, Mongeau M (2012) Conflict free strategic planning. In: Proceeding of the 2012 interdisciplinary science for innovative air traffic management conference. ERAU
Chakravarty A (1985) Four-dimensional fuel-optimal guidance in the presence of winds. J Guid Control Dyn 8(1):16–22
Coppenbarger R, Lanier R, Sweet D, Dorsky S (2004) Design and development of the enroute descent advisor (eda) for conflict-free arrival metering. In: Proceeding of the AIAA-2004-4875 AIAA GNC conference. AIAA GNC
Davis L (1991) Handbook of genetic algorithms. Van Nostrand Reinhold, New York
de Boor C (1978) A practical guide to splines. Springer, New York
Dougui N, Delahaye S, Puechmorel D, Mongeau M (2012) A light-propagation model for aircraft trajectory planning. J Glob Optim 56:873–895
Enright PJ, Conway BA (1992) Discrete approximation to optimal trajectories using direct transcription and nonlinear programming. J Guid Control Dyn 15:994–1002
Erzberger H, Paielli RA, Isaacson DR, Eshowl MM (1997) Conflict detection in the presence of prediction error. In: Proceeding of the air traffic management seminar, FAA/Eurocontrol
Evans J et al (2003) Reducing severe weather delays in congested airspace with weather support for tactical air traffic management. In: Proceeding of the air traffic management seminar, FAA/Eurocontrol
Ewing GM (1969) Calculus of variations with applications. Norton, New York, reprinted by Dover, 1985
Farin G (1993) Curves and surfaces for computer aided geometric design. A practical guide. Academic, San Diego
Farin G, Hansford D (2000) The essentials of CAGD. A K Peters, Natick
Giancoli DC (1989) Physics for scientists and engineers with modern physics, 2nd edn. Prentice-Hall, Englewood Cliffs
Gong Q, Fahroo F, Ross IM (2008) Spectral algorithm for pseudospectral methods in optimal control. J Guid Control Dyn 31(3):460–471
Grimm W, Well K, Oberle H (1986) Periodic control for minimum-fuel aircraft trajectories. J Guid Control Dyn 9(2):169–174
Hargraves CR, Paris SW (1992) Direct trajectory optimization using nonlinear programming and collocation. J Guid Control Dyn 15:994–1002
Heath MT (2002) Scientific computing, an introductory survey. Computer graphics. McGraw-Hill, New York
Jackson MR, Zhao Y, Slattery RA (1999) Sensitivity of trajectory prediction in air traffic management. J Guid Control Dyn 22(2):219–228
Jacobson M, Ringertz UT (2010) Airspace constraints in aircraft emission trajectory optimization. J Aircraft 47:1256–1265
Jain S, Tsiotras P (2008) Multiresolution-based direct trajectory optimization. J Guid Control Dyn 31(5):1424–1436
Jain S, Tsiotras P (2008) Trajectory optimization using multiresolution techniques. J Guid Control Dyn 31(5):1424–1436
Jardin MR, Bryson AE (2001) Neighboring optimal aircraft guidance in winds. J Guid Control Dyn 24:710–715
Jeffreys H, Jeffreys BS (1988) Methods of mathematical physics. Cambridge University Press, Cambridge
Kelley HJ (1973) Control and dynamic systems: advances in theory and applications. Academic, New York
Kirk DB et al (2001) Problem analysis resolution and ranking (parr) development and assessment. In: Proceeding of the air traffic management seminar, FAA/Eurocontrol
LaValle SM (2006) Planning algorithms. Cambridge University Press, Cambridge
Liu W, Hwang I (2012) Probabilistic aircraft mid-air conflict resolution using stochastic optimal control. IEEE Intell Transp Syst Trans Mag
Lu P (1999) Regulation about time-varying trajectories: precision entry guidance illustrated. J Guid Control Dyn 22:784–790
MasalonisA et al (2004) Using probabilistic demand prediction for traffic flow management decision support. In: Proceeding of the AIAA-2004-4875 AIAA GNC conference. AIAA GNC
McNally BD, Bach RE, Chan W (1998) Field test evaluation of the ctas conflict prediction and trial planning capability. In: Proceeding of the AIAA-1998-4480 AIAA GNC conference. AIAA GNC
Meckiff C, Chone R, Nicolaon JP (1998) The tactical load smoother for multi-sector planning. In: Proceeding of the air traffic management seminar, FAA/Eurocontrol
Miele A (1990) Optimal trajectories and guidance trajectories for aircraft flight through windshears. In: Proceedings of the 29th IEEE conference on decision and control. IEEE
Mondoloni S, Bayraktuta I (2005) Impact of factors, conditions and metrics on trajectory prediction accuracy. In: Proceeding of the air traffic management seminar, FAA/Eurocontrol
Mondoloni S, Pagli SM, Green S (2002) Trajectory modeling accuracy for air traffic management decision support tools. In: Proceeding of the ICAS conference. ICAS, Toronto
Oberle HJ, Grimm W (1989) BNDSCO - a program for the numerical solution of optimal control problems. Tech. report. Institute for Flight System Dynamics, German Aerospace Research Establishment Oberpfaffenhofen
Ohtsuka T (2002) Quasi-Newton-type continuation method for nonlinear receding horizon control. J Guid Control Dyn 24:685–692
Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79(1):12–49
Pontryagin LS, Boltyanski VG, Gamkrelidze RV, Mischenko EF (1962) The mathematical theory of optimal processes. Interscience, New York
Pêtrès C, Pailhas Y, Patron P, Petillot Y, Evans J, Lane D (2007) Path planning for autonomous underwater vehicles. IEEE Trans Robot 23(2):331–341
Ramsay JO, Silverman BW (2005) Functional data analysis. Springer series in statistics. Springer, New York
Rao AV, Benson D, Huntington GT (2011) User’s manual for GPOPS version 4.x: a matlab package for software for solving multiple-phase optimal control problems using hp-adaptive pseudospectral methods. Tech. report
Roche E (1997) Parsing with finite-state transducers. In: Roche E, Schabes Y (eds) Finite-state language processing. MIT Press, Cambridge
Ross IM (2005) User’s manual for DIDO: a MATLAB application package for solving optimal control problems. Tech. report, Naval Postgraduate School
Russell RD, Shampine LF (1972) A collocation method for boundary value problems. Numerische Mathematik 19:13–36
Ryan HF, Paglione M, Green S (2004) Review of trajectory accuracy methodology and comparison of error measure metrics. In: Proceedings of the AIAA-2004-4787 AIAA GNC conference. AIAA GNC
Schouwenaars T, How J, Feron E (2004) Receding horizon path planning with implicit safety guarantees. In: American control conference, Boston, MA, June 2004, pp 5576–5581
Schouwenaars T, Valenti M, Feron E, How J, Roche E (2006) Linear programming and language processing for human/unmanned-aerial-vehicle team missions. AIAA J Guid Control Dyn 29(2):303–313
Schultz RL (1990) Three-dimensional trajectory optimization for aircraft. J Guid Control Dyn 13(6):936–943
Schwartz A (1996) Theory and implementation of numerical methods based on runge-kutta integration for solving optimal control problems. Ph.D. thesis, Université Montpellier II, France
Sethian JA (1999) Fast marching methods. SIAM Rev 41(2):199–235
Sethian JA, Vladimirsky A (2003) Ordered upwind methods for static Hamilton-Jacobi equations: theory and algorithms. SIAM J Num Anal 41:325–363
Seywald H (1994) Long flight-time range-optimal aircraft trajectories. J Guid Control Dyn 19(1):242–244
Seywald H, Cliff EM (1994) Neighboring optimal control based feedback law for the advanced launch system. J Guid Control Dyn 17:1154–1162
Seywald H, Cliff EM, Well K (1994) Range optimal trajectories for an aircraft flying in the vertical plane. J Guid Control Dyn 17(2):389–398
Slattery RA, Zhao Y (1997) Trajectory synthesis for air traffic automation. J Guid Control Dyn 20(2):232–238
Sridhar B, Ng HK, Chen NY (2011) Aircraft trajectory optimization and contrails avoidance in the presence of winds. J Guid Control Dyn 34:1577–1583
Strube MJ, Sanner RM, Atkins EM (2004) Dynamic flight guidance recalibration after actuator failure. In: AIAA 1st intelligent systems technical conference. AIAA
SudV et al (2001) Air traffic flow management collaborative routing coordination tools. In: Proceeding of the AIAA-2001-4112 AIAA GNC conference. AIAA GNC
Swensen HN et al (1997) Design and operational evaluation of the traffic management advisor at the forth worth air route traffic control center. In: Proceeding of the air traffic management seminar, FAA/Eurocontrol
Swierstra S, Green S (2003) Common trajectory prediction capability for decision support tools. In: Proceeding of the air traffic management seminar, FAA/Eurocontrol
Tomlin C, Pappas GJ, Sastry S (1998) Conflict resolution for air traffic management: a study in multi-agent hybrid systems. IEEE Trans Automat Control 43:509–521
Vink A (1997) Eatchip medium term conflict detection: part 1 eatchip context. In: Proceeding of the air traffic management seminar, FAA/Eurocontrol
Weiss R (1974) The application of implicit Runge-Kutta and collocation methods to boundary value problems. Math Comput 28:449–464
Williams P (2004) Application of pseudospectral methods for receding horizon control. J Guid Control Dyn 27:310–314
Yan H, Fahroo F, Ross IM (2002) Real-time computation of neighboring optimal control laws. In: AIAA guidance, navigation, and control conference and exhibit. AIAA
Zhao Y (2011) Efficient and robust aircraft landing trajectory optimization. Ph.D. thesis, School of Aerospace Engineering, Georgia Institute of Technology
Zhao Y, Tsiotras P (2010) Density functions for mesh refinement in numerical optimal control. J Guid Control Dyn 34(1):271–277
Zhao Y, Tsiotras P (2010) Time-optimal parameterization of geometric path for fixed-wing aircraft. In: Infotech@Aerospace. AIAA, Atlanta
Zhao Y, Tsiotras P (2011) Stable receding horizon trajectory control for complex environment. In: American control conference (ACC), San Francisco, CA
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Japan
About this paper
Cite this paper
Delahaye, D., Puechmorel, S., Tsiotras, P., Feron, E. (2014). Mathematical Models for Aircraft Trajectory Design: A Survey. In: Air Traffic Management and Systems. Lecture Notes in Electrical Engineering, vol 290. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54475-3_12
Download citation
DOI: https://doi.org/10.1007/978-4-431-54475-3_12
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-54474-6
Online ISBN: 978-4-431-54475-3
eBook Packages: EngineeringEngineering (R0)