Flight trajectory optimization 

Flight trajectory optimization, context and objectives

A detailed description of the concepts, implementation and economical and environmental benefits of performance-based operations, represented by optimal trajectories determined function of the aircraft’s performances and atmospheric conditions is presented by SESAR Joint Undertaking (2015).
Palopo, Windhorst, Suherwardy & Lee (2010) conducted a comparative study of recorded and wind-optimal flight routes, for a period of 365 days, and concluded that the wind-optimal flights yielded an overall economy of 210 lb of fuel, and 2.7 minutes of flight time reduction per flight. Their study showed that these economies were obtained without a significant impact on the airport arrival rates, and reduced the number of conflicts by an average of 29%. In another investigation, on the advantages of using continuous descents at the Atlanta airport, Wilson & Hafner (2005) showed potential flight time reductions of up to 45 hours per day, and fuel cost reductions of 80,000 dollars per day, which amounts to 29 million dollars per year, accompanied by a 50% reduction in the number of potential conflicts.
Dancila, Botez & Ford (2014), and Murrieta Mendoza, Botez & Ford (2016) studied the negative effect of a missed approach (that can be caused, among others, by approach trajectory and runway conflicts, or by adverse atmospheric conditions) on flight time, fuel burn, gas emissions, and costs. These studies show the importance of predicting flight trajectories free of conflicts, especially in the approach and landing phases of the flight. Currently, the flight trajectory optimization is subject to intense research aiming to extend the performances and capabilities of the optimization algorithms by taking advantage of the new, and more powerful computation platforms, new navigation principles and policies, better weather predictions, advancements in aircraft modeling and performance predictions, etc.
A direction of research addressed the lateral and the vertical fight trajectory performance prediction variations due to the differences in platform implementation, as shown in the analysis performed on a set of test benches and simulators from seven different manufacturers by Herdon, Cramer & Nicholson (2009).

Flight trajectory optimization

Huang, Lu & Nan (2012) presented a survey of the numerical algorithms used in aircraft trajectory optimization. Liden (1992a), and Liden (1992b) presented methods for constructing optimal flight profiles that use cruise step climbs, and the effects of the Cost Index (CI) on the minimal cost flight profile, and found two categories of discontinuities for trajectories with Requested Time of Arrival (RTA), that were caused by the modification of the optimum step climb points; Liden (1992a) further proposed approaches for removing these discontinuities.
Bilimoria & Shepard (1989) studied the performances of cruise trajectory optimization using an aircraft range and gross weight state variables–based dynamic model, and three strategies relative to the constant / variable altitude and speed. They also identified the configuration of cruise trajectories as a function of a parameter defining the balance between the aircraft flight time and its fuel burn.
Murrieta Mendoza, Beuze et al. (2015) used a Branch-and-Bound algorithm to determine the optimal speed and altitude profiles defining the still-air optimal vertical flight trajectory. This trajectory had minimized the total flight cost of an aircraft whose performance model was described using look-up tables.
Bonami, Olivares, Soler & Staffeti (2013) investigated an aircraft trajectory optimization algorithm based on mixed integer non-linear programming, by using continuous and discrete variables representing aircraft state and decision variables, respectively. The algorithm was implemented using Gauss-Lobatto direct collocation and Branch-and-Bound algorithms.
In their paper, Tong, Schoemig, Boyle, Scharl & Haraldsdottir (2007) presented a set of factors related to the selection of a ground-referenced geometric path for a Continuous Descent and Approach (CDA), and the compromises entailed by low and idle thrust descents. Their study analyzed the characteristics of idle and constant path angle profiles for two aircraft types (B777-200 and B737-700), and a number of aircraft descent configurations. They showed that while an idle descent provided the best performances relative to fuel and gas emissions, its ground-referenced path could not be predicted due to variations in aircraft performances, configurations, and wind conditions. On the other hand, descents following a predetermined geometric path, requiring low thrust were predictable and, thus, usable in a high traffic ATC environment.

Opportunities of improvement for the trajectory optimization algorithms

The reviewed literature showed the strong interest in the exploration of new optimization algorithms, and the wide range of trajectory optimization algorithms developed to date. The review also emphasized the importance of the support algorithms used by the optimization process such as the selection of the set of candidate waypoints, the aggregation of the necessary weather information, and the aircraft performance prediction computations. The capabilities of the support algorithms (such as the position and number of selected candidate waypoints, the accuracy of the weather information, or the speed and accuracy of the aircraft performance predictions) have a determining contribution to the performances (speed and accuracy) of the optimization algorithms. These observations led to the interest to also investigate new methods for improving the accuracy and capabilities of the support algorithms.
A first area that warranted further investigation was the selection of the candidate waypoints considered in the process of optimization. As shown by Gil (2011), the selection of a set of candidate waypoints disposed as a rhomboidal grid may lead to algorithm inefficiencies due to the evaluation of waypoint that will never be retained in the final trajectory. Other selection methods such as waypoint grids built based on the OTS, investigated by Rodionova et al. (2014), or on a set of parallel tracks situated on both sides of the geodesic (Félix Patrón, Kessaci et al., 2013) may lead to the exploration of geographic areas that do not take full advantage of the favorable wind conditions. These methods, too, may lead to optimization algorithm inefficiencies. The solution presented by Devulapalli (2012) allowed a better adaptation of the grid to the position of the departure and destination points, and to aircraft turn performances. However, upon an in-depth review regarding the construction of the ellipse, it could be observed that the proposed method had an important drawback – directly linking the shape of the ellipse and the grid, controlled by the eccentricity, to the aircraft’s turn performance and terminal nodes’ heading constraints. Depending on the airports’ positions and the aircraft’s turn performances, this fact may also lead to the selection of geographic areas which could be too wide (including candidate nodes situated too far, thus, not part of the solution) or too narrow (excluding geographic areas and candidate nodes that could be part of the optimal solution). Therefore, this method may also lead to computation inefficiencies.A second area that warranted further investigation was represented by the aircraft performance modeling and aircraft trajectory prediction methods. Irrespective of the optimization method used, a solution is selected following an iterative computation and evaluation of a series of trajectories or trajectory segments’ performances. Moreover, the optimization algorithm may be executed at regular time intervals in order to update the optimal trajectory as function of the evolution of the wind conditions, aircraft position and its state parameters. Therefore, an optimization algorithm could be improved by employing new aircraft performance calculation and/or trajectory prediction methods that would provide more accurate trajectory predictions, allow the evaluation of more candidate optimal trajectories in a given time frame, and / or ensure faster optimization algorithms’ response times. An example of such algorithm was presented by Dancila (2011) and Dancila et al.(2013).
A third area of interest was the trajectory optimization itself. More specifically, the investigation of a new method addressing the geometrical optimization of the vertical flight plan, having as principal objectives the reduction of the flight plan computation complexity and time. This method would facilitate the trajectory optimization when the succession of waypoints is pre-imposed. As well, this method would provide a fast and straightforward way for computing a full lateral and vertical flight plan when the set of waypoints is not imposed (such as the case with general trajectory optimization algorithms).

Area selection on an ellipsoid

For any and all aircraft navigation and flight trajectory prediction/optimization computations, the set of waypoints composing the flight trajectory is situated on and defined with respect to the surface of the earth. Therefore, on the reference ellipsoid, the desired contour corresponds to that of the selected surface-plane ellipse which is subsequently ‘‘molded’’ on the ellipsoid such that P1 and P2 map to their corresponding geographical coordinates by using a process known as conformal mapping (Leick, 1985; Grafarend & Krumm, 2006; Kayton & Fried, 1997; Deakin, 2004; Agard & Gehring, 1965). Consequently, the surface-plane ellipse’s P1P2 segment and its major axis translate into the orthodrome between P1 and P2, and a segment of the ellipsoid’s grand circle passing through P1 and P2, respectively. Also, the center of the surface-plane ellipse translates into a point situated on the orthodrome, at mid-distance between P1 and P2. The heading at each point along the orthodrome is not constant, thus it changes continuously (Karney, 2013). The orthodrome’s heading at the mid-point C between P1 and P2 represents the surface-plane ellipse’s rotation angle relative to the meridian passing through C, prior to its molding onto the ellipsoid (Leick, 1985).
Consequently, the ‘‘geographical area selection algorithm’’ must determine the geographical coordinates of a set of points situated on the surface of the reference ellipsoid that correspond to the contour of an ellipse which satisfies the set of pre-defined criteria:
its major axis is centered on the orthodrome linking the two airports – thus including the trajectory of minimal total ground distance and ensuring a symmetrical exploration of the area wind conditions;
it ensures a prescribed maximal total ground distance between the two airports; and it provides a specified range around the airports for take-off and landing maneuvers.

Construction of the routing grid

The routing grid is constructed using the specified latitude and longitude axial resolution required by the flight trajectory computation or optimization algorithm; it circumscribes the contour delimiting the selected geographical area.
Depending on the selected ellipse’s major axis length, eccentricity, center position and tilt angle, the shape of the orthodrome, and by consequence the shape of the contour delimiting the selected geographical area with respect to the ellipsoid surface’s reference coordinate system represented in a geographic projection (a Cartesian, Latitude and Longitude projection) can be concave, convex or symmetric. It can be noted that irrespective of the contour’ shape, a maximum of two contour points can exist at each longitude. On the contrary, depending on the ellipse’s major axis length, eccentricity, center position and tilt angle, the contour of the selected geographical area can take a strong concave shape which makes that the number of contour points corresponding to a selected latitude could reach a value of four. Moreover, on its orthodromic flight from the departure to the destination airport, the aircraft’s longitude variation is generally uniform (with exceptions related to take-off and landing procedure maneuverings), whereas its latitude variation may not be. Therefore, it seemed appropriate to organize the grid data following the longitude, i.e. the range of longitudes covered by the selected geographical contour and the corresponding range of latitudes for each longitude. The selected contour’s minimal and maximal longitude and latitude values are identified and subsequently used in conjunction with the grid’s required resolution for determining the sets of latitude and longitude values specific to the desired grid. By travelling the contour in a clock-wise direction, starting with the point situated on the top (northern) side at the longitude corresponding to the center of the ellipse, the algorithm determines the corresponding grid point coordinates (longitude and latitude) by performing an ‘‘upward’’ or ‘‘downward’’ rounding.

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Table des matières

1.1 Flight trajectory optimization, context and objectives
1.2 Flight trajectory optimization
1.3 Opportunities of improvement for the trajectory optimization algorithms
3.1 Introduction
3.2 Geographical area selection
3.2.1 Ellipse area selection on a plane surface
3.2.2 Area selection on an ellipsoid
3.2.3 Implementation
3.3 Construction of the routing grid
3.4 Results
3.5 Conclusions
4.1 Introduction
4.2 Existing algorithms’ vertical and lateral flight path segment parameters’ computation
4.3 Description of the proposed method
4.3.1 Input configuration data
4.3.2 Gross weight and center of gravity position
4.3.3 Maximum flying altitude as function of the gross weight
4.3.4 The climb flight path and the Top Of Climb
4.3.5 Descent flight paths and the set of Top of Descent points
4.3.6 Cruise vertical flight paths Step-climb vertical flight paths Level-flight cruise vertical flight paths
4.3.7 The vertical flight path look-up structure and the vertical flight path graph
4.3.8 The vertical and lateral flight plan computation using the vertical flight plan look-up structure and the vertical flight plan graph
4.4 Results
4.5 Conclusions
5.1 Introduction
5.2 The Optimization Methodology
5.2.1 Pre-processing
5.2.2 Vertical Flight Plan Optimization
5.3 Results
5.4 Conclusion

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