Spline parameterization based nonlinear trajectory optimization along 4D waypoints

  • Ahmed, Kawser (LAETA/UBI - AeroG, Laboratory of Avionics and Control, Department of Aerospace Sciences, University of Beira Interior) ;
  • Bousson, Kouamana (LAETA/UBI - AeroG, Laboratory of Avionics and Control, Department of Aerospace Sciences, University of Beira Interior) ;
  • Coelho, Milca de Freitas (LAETA/UBI - AeroG, Laboratory of Avionics and Control, Department of Aerospace Sciences, University of Beira Interior)
  • Received : 2018.10.02
  • Accepted : 2019.05.08
  • Published : 2019.09.25


Flight trajectory optimization has become an important factor not only to reduce the operational costs (e.g.,, fuel and time related costs) of the airliners but also to reduce the environmental impact (e.g.,, emissions, contrails and noise etc.) caused by the airliners. So far, these factors have been dealt with in the context of 2D and 3D trajectory optimization, which are no longer efficient. Presently, the 4D trajectory optimization is required in order to cope with the current air traffic management (ATM). This study deals with a cubic spline approximation method for solving 4D trajectory optimization problem (TOP). The state vector, its time derivative and control vector are parameterized using cubic spline interpolation (CSI). Consequently, the objective function and constraints are expressed as functions of the value of state and control at the temporal nodes, this representation transforms the TOP into nonlinear programming problem (NLP). The proposed method is successfully applied to the generation of a minimum length optimal trajectories along 4D waypoints, where the method generated smooth 4D optimal trajectories with very accurate results.


trajectory optimization;4D waypoint navigation;spline parameterization;Nonlinear programming


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