Open Access Open Access  Restricted Access Subscription or Fee Access

Generating Time Optimal Trajectory from Predefined 4D waypoint Networks

Kawser Ahmed(1), Kouamana Bousson(2*)

(1) Avionics & Control Laboratory, Department of Aerospace sciences, University of Beira Interior, Portugal
(2) Avionics & Control Laboratory, Department of Aerospace sciences, University of Beira Interior, Portugal
(*) Corresponding author



The main purpose of this paper is to develop a trajectory optimization method to generate optimal trajectories that minimize aircraft total trip time between the initial and final waypoint in predefined 4D waypoint networks. In this paper, the 4D waypoint networks only consist of waypoints for climb, cruise and descent approach without the take-off and landing approach phases. The time optimal trajectory is generated for three different lengths of flights (short, medium, and long-haul flight) for two different commercial aircraft and considering zero wind condition. The Results about the presented applications show that by flying a time optimal trajectory, which was found by applying a single source shortest path algorithm (Dijkstra’s algorithm), can lead to the reduction of average travel time by 2.6% with respect to the total trip time.
Copyright © 2017 Praise Worthy Prize - All rights reserved.


Time Conservation; Cost Index; 4D Waypoint Navigation; Trajectory Optimization; Dijkstra’s Algorithm; Base of Aircraft Data (BADA)

Full Text:



Roberson B, "Fuel Conservation Strategies: Cost Index Explained," Boeing, 2007, pp. 26-28.

Bryson A. E and Ho Y. C, "Applied Optimal Control: Optimization, Estimation and, Control", Taylor & Francis, New York, 1975.

Betts, J.T., “Survey of numerical methods for trajectory optimization” Journal of Guidance, Control and Dynamics, 1998, pp. 193-207.

Pontryagin, L.S., V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko. The Mathematical Theory of Optimal Processes, Wiley-Interscience, New-York, 1962.

Von Stryk, O. and R. Bulirsch. “Direct and indirect methods for trajectory optimization”, Annals of Operations Research, 37, 1992, pp. 357-373.

Hull, D.G., “Conversion of Optimal Control Problems into Parameter Optimization Problems”, Journal of Guidance, Control, and Dynamics, 1997, pp. 57-60.

Schwartz, A. and E. Polak, “Consistent approximations for Optimal Control Problems Based on Runge-Kutta Integration”, SIAM Journal on Control and optimization, vol.34, No.4, 1996, pp. 1235-1269.

Hargraves, C.R. and S.W. Paris, “Direct Trajectory Optimization Using Nonlinear Programming and Collocation”, Journal of Guidance, Control and Dynamics, vol. 10, No.4, 1987, pp. 338-342.

Bousson, K. “Chebyshev pseudospectral trajectory optimization of differential inclusion models”, SAE World Aviation Congress, Montreal, Canada, paper no. 2003-01-3044, 2003.

Fahroo, F. and I.M. Ross, “Direct trajectory optimization by a Chebyshev pseudospectral method”, Journal of Guidance, Control and Dynamics, 2002, pp. 160-166.

Bousson K. and Machado P, "4D Flight Trajectory Generation and Tracking for Waypoint-Based Aerial Navigation," WSEAS transactions on system and control, Vol 8, No 3, july 2013, pp 105-119.

Boukraa, D., Bestaoui, Y., Azouz, N., Three Dimensional Trajectory Generation for an Autonomous Plane, (2014) International Journal on Numerical and Analytical Methods in Engineering (IRENA), 2 (4), pp. 144-154.

Bousson, K., Gameiro, T., A Quintic Spline Approach to 4D Trajectory Generation for Unmanned Aerial Vehicles, (2015) International Review of Aerospace Engineering (IREASE), 8 (1), pp. 1-9.

Devika, K., Thomas, S., Path Planning for Conflict Resolution in Free Flight System: Optimization Based on Linear Programming, (2016) International Review of Aerospace Engineering (IREASE), 9 (1), pp. 1-6.

Cormen T. H, Leiserson C. E, Rivest R. L and Stein C, "Introduction to algorithms", London, England: The MIT press, 2009, pp. 658-659.

Hart C, "Graph Theory Topics in Computer Networking," (2013, pp 13-20).

Dasgupta S, Papadimitriou C. H and Vazirani U. V, "Algorithms", McGraw-Hill, New York, July 18, 2006, pp. 112-118.

Seemkooei A. A, "Comparison of different algorithm to transform geocentric to geodetic coordinates," Survey Review 36, October 2002, pp 627-632.

EUROCONTROL "Aircraft Performance Summary Tables for the Base of Aircraft Data (BADA)," Eurocontrol Experimental Centre, Revision 3.4, June 2002.

"SkyVector," Website Available: [Accessed October 2015].


  • There are currently no refbacks.

Please send any question about this web site to
Copyright © 2005-2021 Praise Worthy Prize