The current lack of efficiency in the use of airspace, enforced by the economic crisis an the increasing competitiveness among air transport companies are drivers for a renewed interest for the application of optimization techniques to find answers to overcame current inefficiencies.The main objective of the present paper is to assess and compare different optimal aircraft trajectories techniques applied to the minimum fuel cruise problem at constant altitude and course with fixed arrival time, International Standard Atmosphere and without wind. Four trajectory optimization methods have been used: Hermite-Simpson, 5th degree Gauss-Lobatto and Radau pseudospectral collocation methods and the singular arc solution.Hermite-Simpson and 5th degree methods have been programmed in Ampl modeling language with an IPOPT solver and Radau pseudospectral method using gpops matlab tool with SNOPT solver.5th degree Gauss-Lobatto collocation method gives the less fuel consumption solution followed by Radau pseudospectral, Hermite-Simpson and singular arc. In considering the program execution time, Hermite-Simpson collocation method is the fastest method followed by 5th degree and Radau pseudospectral. Also, taking into account the time for developing the program code the Radau pseudospectral is the most user friendly. Moreover, it has been observed that increasing the sample points in the Hermite- Simpson and 5th degree, the solution converge to the minimum fuel consumption solution. On the other hand, gpops does not show much sensitivity to the number of sample points

In this study, the environmental realities of enhancing departure aircraft procedures, based on optimal Continuous Climb Operations (CCOs), are discussed for a long-range wide-body aircraft. The findings are driven by the advantage offered by the analysis of real Flight Data Recorder (FDR) data against the simulated data, obtained through the use of orthogonal polynomials such as Chebyshev polynomials, in partnership with Gauss-type integration rules. Therein, the calculation of improved trajectories aims at reducing the negative environmental aspects of civil aviation on urbanized areas around airports and enhancing the operational cost efficiency. The multi-objective trajectory optimization has not only been applied to multiple environmental factors particularly; fuel consumption and noise impacts, but also has been constrained by Air Traffic Control (ATC) operational restrictions from a real scenario. The problem, tackled through a multi-objective optimization process based on CCO principles by a Chebyshev–Gauss–Lobatto (CGL) Pseudospectral Method, relies on a numerical framework that bears out the benefits in terms of fuel consumption and noise emissions. Therefore, this study promotes the need for investigating, not the implementation of a CCO generally assumed as an uninterrupted climb departure, but an optimal CCO.

In the future air traffic management system, the trajectory becomes the fundamental element of a new set of operating procedures collectively referred to as trajectory-based operations. Trajectory-based operations require the air traffic management to introduce profound innovations to enable the envisioned changes. Some of these include collaborative decision-making processes, better data and information management, and advanced decision support tools to aid human operators. In particular, fast and accurate computation of optimal trajectories could certainly contribute to enhance trajectory management within the future air traffic management. The trajectory optimization problem can be solved using optimal control methods. In this paper, the existing methods for solving optimal control problems focusing on direct collocation are discussed. In particular, pseudospectral collocation methods have shown to be numerically more accurate and computationally much faster than other direct methods. A very relevant problem arising in air traffic management is analyzed, that is the minimum-fuel trajectory with required time of arrival. Preliminary results illustrate the advantages of using pseudoespectral collocation methods for trajectory optimization in air traffic management; for the same order of numerical accuracy, they present computational times that are at least two orders of magnitude better than other collocation methods. For similar computational times, pseudospectral collocation methods achieve a numerical accuracy that is at least one order of magnitude better than other methods.