Fig. 1.1. Potential contribution of different measures to the reduction of the net CO2 emissions of international aviation. Source: IATA [IATA, 2013].
The aviation sector represents a major challenge to the environment, being one of the top ten emitters of greenhouse gases (GHG) in the world, and accounting for approximately 4% of human contribution to the global warming [Klower et al., 2021, Lee et al., 2009]. According to a recent report by the intergovernmental panel on climate change (IPCC) [IPCC, 2018], a leading climate science body, aviation is a crucial sector to the threat of climate change and global warming. Indeed, although air transport is only responsible for 14% of the transport sector’s carbon dioxide emissions, far less than other modes of transport like light-duty vehicles (30%) and heavy-duty vehicles (36%), it seems to be harder to decarbonize. The reason for this lies in the predicted growth rate for air traffic, which is projected to be greater than for other modes of transport.
The predicted growth rate for air traffic is another concerning aspect. Before the COVID-19 crisis, the compound annual growth rate (CAGR) forecast over the next two decades, assessed by the international air transport association (IATA), estimated that the number of flights was expected to continue increasing by 3.5% annually, which translates to twice the current number of passengers by 2040 [IATA, 2016]. It is well known that the COVID-19 pandemic has severely affected the whole aviation sector, international commercial flights in particular, and, consequently, air traffic has plummeted by more than two thirds compared with previous levels.1 At present, vaccination rates, travel restrictions, and quarantine procedures are rendering it unclear how quickly the aviation sector will return to normal values. However, air traffic growth is expected to recover to pre-crisis levels within the next few years, while continuing to increase annually [Lai et al., 2022]. This continuous growth in air traffic demand imposes the search for new solutions to mitigate its potential negative effects on our planet.
Unfortunately, the measures taken by the aviation sector until recently are considered to be insufficient to mitigate its impact on climate change. Current policies should be strengthened and a firm commitment from all the stakeholders is essential to success. For instance, the IATA is looking for improvements in different areas in order to achieve a reduction in net aviation CO2 emissions of 50% by 2050, relative to 2005 levels [IATA, 2013]. The different measures considered can be classified into technical improvements, including more efficient aircraft and engines, operational measures, including more efficient flight procedures and airport operations, and the search of alternative, more sustainable fuels [Abrantes et al., 2021]. Fig. 1.1 gives a schematic overview of the potential contribution of the mentioned measures to the reduction of the net CO2 emissions of international aviation.
Fig. 1.1. Potential contribution of different measures to the reduction of the net CO2 emissions of international aviation. Source: IATA [IATA, 2013].
As such, the predicted effects of the aviation sector on climate change are greater than desired [IPCC, 2018, ICAO, 2016], and a great effort is required to mitigate them. In this regard, formation flight has the potential to make a significant contribution. Furthermore, formation flight offers great promise not only with regard to mitigating the environmental impact of aviation but also in terms of increasing air traffic management (ATM) capacity and reducing air traffic control workload, which represents another major challenge of the sector.
Formation flight is where two or more aircraft fly at a short distance from each other. The notion of formation flight arose from the observation of nature, in particular from the different species of bird that fly in formation during their migrations [Lissaman and Shollenberger, 1970]. Aircraft formation flight was first introduced in military aviation for mutual defense during missions, concentrating the fire power during attacks, and, consequently, improving combat efficiency. Later, it was demonstrated that formation flight led to notable efficiency improvements for the birds and that substantial fuel savings were also possible for aircraft flying in formation. The reduction achieved strongly depends on the longitudinal distance between aircraft, this being negligible over approximately 40 wingspans. Formations in which aircraft fly at a longitudinal distance between 10 and 40 wingspans are called extended formations [Durango et al., 2016]. They represent a trade-off between safety and efficiency. This thesis will investigate the possibility of implementing extended formations in commercial aviation. In particular, the focus is on the possibility of implementing economically and environmentally beneficial formation flights in the presence of different sources of uncertainty, namely the departure times of the flights and the fuel savings during formation flight.
Fig. 1.2. Two civil aircraft in formation flight. Source: Airbus2
The key enabling factors for this concept of operation consist of a formation control system to keep the aircraft flying in formation at the optimal relative position to optimize the fuel burn savings for the trailing aircraft and an ATM system capable of synchronizing flights departing from different airports to ensure that the formation mission occurs as planned. Within a formation flight, instabilities such as meandering and external factors may affect the motion of the vortices. It is therefore important to maintain the relative position between the follower aircraft and the leader’s wake vortices precisely, as the fuel burn savings are very sensitive to that relative positioning. This can be done using a formation control system capable of continuously locating the wake vortices, maintaining the aircraft in the optimal relative position, and, in this way, optimizing the fuel savings during the formation flight [Caprace et al., 2019]. While technical issues related to maintaining an efficient and safe formation flight have been solved in recent years, some concerns have yet to be addressed. In particular, a major change in airworthiness standards, policies, and procedures is required. At the 40th international civil aviation organization (ICAO) Assembly, formation flight was proposed as a strategic objective, and the necessity of developing a new operational concept including reduced separations between aircraft to allow formation flights was established [Abeyratne, 2020].
Due to its potential for reducing fuel consumption and GHG emissions, formation flight has recently attracted increasing interest, and the studies on the aerodynamics of formation flight have given way to studies on formation flight planning. The majority of the commercial formation flight planning studies are based on geometric models which do not allow use of either accurate dynamic models or meteorological forecasts. The past few years have brought with them some research studies [Hartjes et al., 2018, Hartjes et al., 2019] in which the mission design problem has been studied for two- and three-aircraft formations using a multiphase optimal control approach. This methodology has clear advantages compared to previous ones since the optimal control approach allows the use of accurate models, improving the predictability of the trajectories. However, using the multiphase approach implies having a fixed switching structure in advance, that is to say, knowing the transition among phases established in advance. Since the switching structure of the optimal solution is unknown for the formation mission design problem, any combination should be addressed and, then, the obtained solutions must be compared to establish which one corresponds to the minimum operation-related aircraft costs, also known as direct operating costs (DOC) [Camilleri, 2018]. Thus, this approach is feasible if the number of flights considered is low and both the type of formation and the relative position of each aircraft in the formation are fixed. Otherwise, the number of combinations quickly increases with the number of flights, making the previous approach hardly scalable.
A more general framework is therefore needed, which is able to design formation missions solving a single optimal control problem (OCP) in which neither the number of phases nor the switching structure is known in advance. Moreover, this approach should be easily scalable to design, for instance, formation missions involving more than three flights. For this purpose, the formation mission design problem can be formulated as an OCP of a switched dynamical system. These systems are described by both continuous and discrete dynamics in which the transitions among discrete states are not established in advance. In particular, each aircraft has different flight modes, namely solo flight and flight in different positions within a formation, and their combination is represented by the discrete state of the switched dynamical system, which models their joint dynamic behavior. Each flight mode will be represented by different dynamical equations, which may or may not include formation flight benefits in terms of fuel savings. Additionally, logical constraints in disjunctive form, based on the streamwise distance between aircraft, model the switching logic among the discrete states of the system. To solve the proposed problem, the embedding technique [Bengea et al., 2011] has been employed to obtain a new smooth formulation of the problem in which the switching dynamics are defined without binary variables together with equality and inequality constraints. The obtained problem can be solved using classical optimal control techniques.
In addition to using accurate models, the operational concept of the formation flight entails a deeper understanding of how uncertainties affect the trajectories of each aircraft involved in the formation. In particular, addressing the uncertainty quantification in formation flight design problems is essential to improving the predictability of the trajectories and achieving more realistic solutions and estimations of the costs and the formation benefits. There are different sources of uncertainties which can affect the whole ATM system. One of the main ones is uncertainty in flight departure times, which causes trajectory uncertainty and ATM-context inefficiencies [Rivas and Vazquez, 2016]. Timing is also a crucial factor in formation missions. Effectively, considering the usual cruise speed of most long-haul commercial aircraft, missing the rendezvous location by, for instance, ten minutes means spatially missing the partner aircraft by 150 km. Such cases require catch-up maneuvers, which result in a loss of performance compared to the planned formation mission. As such, uncertainties in departure times have been considered in the formation mission design problem, along with uncertainties in the fuel savings during formation flight. These uncertain parameters have been modeled as random variables described by probability density functions.
The stochastic formation flight mission design problem can be formulated as follows. Given several commercial flights together with the relevant weather forecast, the problem consists in establishing how to organize them in formation, of two- or three-aircraft, or solo flights and in finding the trajectories that minimize the expected DOC value of the formation mission. To solve it, the formation mission design problem is formulated as an OCP of a stochastic switched dynamical system and solved using nonintrusive generalized polynomial chaos (gPC) based stochastic collocation [Matsuno et al., 2015, Li et al., 2014]. The gPC method converts the stochastic switched OCP into an augmented deterministic switched OCP, which can be solved using the embedding approach together with classical numerical optimal control techniques. This technique allows statistical and global sensitivity analysis of the stochastic solutions to be conducted at a low computational cost.
The solution of the formation mission design problem in the presence of uncertainties is stochastic, i.e, its components are random processes, which can be characterized by their mean and standard deviation functions. The expected values and the standard deviations of the latitude and longitude of the optimal trajectories with respect to the expected value of the timing and the expected values and standard deviations of the timing of the trajectory as functions of the expected value of the distance are useful for ATM purposes such as conflict detection and traffic synchronization. The expected values and standard deviations of the fuel consumption and final time of each flight of the formation mission permit the DOC to be estimated in such a way that airlines can establish to what extent a formation mission is economically beneficial.
The relative contributions of each random variable to the variability of the latitude and longitude of the optimal trajectories of each aircraft as functions of the expected value of timing and the variability of the timing of the trajectories of each aircraft as functions of the expected value of the distance allow both the ATM staff to establish which sources of uncertainty must be reduced to increase the predictability of the trajectories and airlines to determine what sources of uncertainty must be reduced to decrease the DOC.
It is well known that uncertainty regarding the weather forecast, in particular the wind field, is of particular interest in the context of optimization trajectories. However, the stochastic impact of the wind field in the formation mission design problem has not been considered. Being a random function of space and time, a wind field should be modeled as a stochastic process. Although generalized polynomial chaos can be employed to represent stochastic processes, specific techniques are needed for their representation, such as the Karhunen—Loeve expansion. As a consequence, the dimensionality of the problem increases, as does the complexity of dealing with it. Therefore, it is assumed that the aircraft model is accurately known, as is the corresponding weather forecast. Neither storms nor operational uncertainties have been considered. The study of the formation mission design problem in the presence of stochastic processes such as uncertain wind field, is left for future research. In this thesis, only random variables have been considered.
Recently, the aviation industry has shown renewed interest in achieving the real implementation of commercial formation flight. Indeed, during the completion of this thesis, the Airbus Fello’fly project [Airbus, b] has been carried out, in which Airbus collaborated with airlines, air navigation service providers, and civil aviation authorities to tackle the challenges of formation flight and demonstrate its operational feasibility. The first flight test in the field of transoceanic flights was conducted in November 2021.3 This test, in which a system for formation flight control proved itself in an operational environment, shows that this technology has reached the highest readiness level, according to the European Union’s scale [Héder, 2017].
The general objective of this thesis is to solve the formation mission design problem for commercial aircraft in the presence of uncertainties.
The stochastic formation mission planner should include accurate dynamic models of the aircraft involved in the formation mission, the relevant flight data such as departure and arrival times and locations, wind forecast, a model of the effects of flying in formation on the fuel consumption of the trailing aircraft, the probability density functions that characterize the uncertain parameters of the problem, and an objective functional reflecting the DOC of the formation mission.
The stochastic formation mission planner presented in this thesis, which is based on stochastic optimal control, finds the optimal solution for up to three aircraft candidates to fly in formation, establishing how to arrange them in formation or solo and determining the stochastic optimal trajectories. The stochastic solution of the formation mission planning problem includes the expected values and standard deviations of the latitude, longitude, and timing of the optimal trajectories. The solution also includes the expected values and standard deviations of the latitude, longitude, and timing of the rendezvous and splitting points as well as the expected values and standard deviations of the arrival times of the flights. Finally, by performing the sensitivity analysis of the solutions, the relative influence of the uncertain parameters of the problem on the variability of the stochastic solution components can be estimated. The schematic representation of the elements of the formation mission planner is given in Fig. 1.3.
The general objective of this thesis has been divided into several specific objectives:
•Reviewing existing literature on formation mission planning.
•Developing accurate dynamic models of commercial aircraft.
•Devising a method for incorporating the wind forecast into an optimal control problem.
•Formulating the deterministic formation mission design problem as an OCP of a switched dynamical system.
•Developing appropriate optimal control techniques to solve the deterministic formation mission design problem.
•Formulating the stochastic formation mission design problem as a stochastic OCP of a switched dynamical system.
•Developing the appropriate optimal control techniques to solve the stochastic formation mission design problem.
•Numerically solving several instances of the stochastic formation mission design problem in the presence of uncertainties regarding aircraft departure times and the fuel savings during formation flight to demonstrate the effectiveness of the method.
•Performing sensitivity analyses of the stochastic solution to estimate the relative influence of the uncertain parameters of the problem on the variability of the solution components.
Fig. 1.3. Schematic representation of the elements of the stochastic formation mission planner presented in this thesis.
The solutions obtained demonstrate that formation flight is environmentally and economically beneficial in the presence of uncertainties regarding the departure times of the aircraft currently observed at major international airports and the fuel savings achieved during formation flight.
The formation mission planner presented in this thesis is intended to be incorporated into a ground-based system which could be used by flight dispatchers as a support system to design formation missions for airlines or airline alliances. Flight dispatchers are employed by airlines who are responsible for compiling and modifying the flight plan. They also monitor the aircraft during flight. The computational time of the proposed algorithm makes it suitable for the strategic and tactical planning phases of the flight, as well as for the operational phase, in other words, during flight.
The contributions of this thesis to the state of the art in formation mission planning can be summarized as follows:
Contribution A: First, an innovative approach to the deterministic formation mission design problem is proposed which is formulated as an optimal problem of a switched dynamical system with logical constraints in disjunctive form and solved using an embedding approach. The optimal control formulation of the problem allows accurate dynamic models of aircraft and meteorological forecasts to be included in the problem formulation in order to improve the predictability of the trajectories and the estimation of the fuel savings. Modeling the formation mission as a switched dynamical system allows previous approaches such as the multiphase approach, to be avoided. The embedding method employed to solve the resulting OCP of a switched dynamical system is able to deal with both the switching dynamics of the system and the logical constraints in disjunctive form in an efficient way. Additionally, the switching logic among discrete states can be modeled without using binary variables, the number of switches does not have to be established in advance, and the optimal values of the switching times between discrete states are obtained without introducing them as unknowns of the OCP. Therefore, the resulting problem is a classical OCP, which has been solved using a pseudospectral knotting method leading to significant reductions in the computational time with respect to previous approaches.
Contribution B: Then, considering the presence of uncertainties in the problem, the methodology for the solution of the stochastic formation mission design problem has been introduced, which is formulated as an OCP of a stochastic switched system. The uncertainties are represented by random variables characterized by probability density functions. This problem is solved using an approach based on gPC, in which the stochastic switched OCP is transformed into an augmented deterministic switched OCP in a higher dimensional state space. This problem is thus solved using the same methodology described in Contribution A. Several numerical experiments have been conducted in which uncertainties regarding the departure times and the fuel saving during formation flight have been considered, demonstrating its effectiveness.
Contribution C: Finally, the methodology used to solve the formation mission design problem in the presence of uncertainties makes it possible to conduct statistical and global sensitivity analysis of the stochastic solution at a low computational cost. The statistic analysis consists in the computation of the expected values and standard deviations of the geographical coordinates and timing of the aircraft trajectories obtained in the stochastic solution. The purpose of the sensitivity analysis is to estimate the relative contribution of the random parameters to the variability of the components of the stochastic solution with the aim of reducing its variability acting on the source of uncertainty.
The rest of the document is organized as follows. In Chapter 2, an overview of formation flight is given. In Chapter 3, the model of the system that represents the formation flight is introduced. In Chapter 4, the optimal control method employed to solve the deterministic formation mission design problem is introduced. In Chapter 5, a general overview to numerical methods for optimal control is given. In Chapter 6, the stochastic optimal control method employed to solve the stochastic formation mission design problem is introduced. In Chapter 7, the methodology introduced in Chapter 4 is employed to solve three different formation mission design problems in the absence of uncertainties. In Chapter 8, the methodology introduced in Chapter 6 is employed to solve two different formation mission design problems in the presence of uncertainties. Finally, the conclusions and a discussion of the future work are outlined in Chapter 9.
[Abeyratne, 2020] Abeyratne, R. (2020). The outcome of the 40th ICAO assembly: A new look at ICAO? Air and Space Law, 45(1).
[Abrantes et al., 2021] Abrantes, I., Ferreira, A. F., Silva, A., and Costa, M. (2021). Sustainable aviation fuels and imminent technologies-CO2 emissions evolution towards 2050. Journal of Cleaner Production, 313:127937.
[Airbus, a] Airbus. https://www.airbus.com/en/newsroom/press-releases/2021-11-airbus-and-its-partners-demonstrate-how-sharing-the-skies-can-save. Accessed: 2021-11-18.
[Airbus, b] Airbus. https://www.airbus.com/en/innovation/disruptive-concepts/biomimicry/fellofly. Accessed: 2022-01-10.
[Alwan and Liu, 2018] Alwan, M. S. and Liu, X. (2018). Theory of Hybrid Systems: Deterministic and Stochastic. Springer.
[Ansari et al., 2020] Ansari, M. Y., Ahmad, A., Khan, S. S., Bhushan, G., et al. (2020). Spatiotemporal clustering: A review. Artificial Intelligence Review, 53(4):2381–2423.
[Assembly 40th, 2019] Assembly 40th, I. https://www.icao.int/Meetings/a40/Documents/WP/wp_317_en.pdf. Accessed: 2021-11-21.
[Aung and Tan, 2010] Aung, H. H. and Tan, K.-L. (2010). Discovery of evolving convoys. In Gertz, M. and Ludäscher, B., editors, International Conference on Scientific and Statistical Database Management, pages 196–213. Springer.
[Bengea et al., 2011] Bengea, S., Uthaichana, K., Zěfran, M., and DeCarlo, R. A. (2011). Optimal control of switching systems via embedding into continuous optimal control problem. In Levine, W. S., editor, The Control Handbook: Advanced Methods, chapter 31, pages 31–1 – 31–23. CRC Press.
[Bengea and DeCarlo, 2005] Bengea, S. C. and DeCarlo, R. A. (2005). Optimal control of switching systems. Automatica, 41:11–27.
[Betts, 2010] Betts, J. T. (2010). Practical Methods for Optimal Control and Estimation using Nonlinear Programming. SIAM.
[Bieniawski et al., 2014] Bieniawski, S. R., Clark, R. W., Rosenzweig, S. E., and Blake, W. E. (2014). Summary of flight testing and results for the formation flight for aerodynamic benefit program. In Proceedings of the 52nd Aerospace Sciences Meeting, National Harbor, MD, USA.
[Blake and Flanzer, 2016] Blake, W. B. and Flanzer, T. C. (2016). Optimal routing for dragreducing formation flight: A restricted case. Journal of Guidance, Control, and Dynamics, 39(1):173–176.
[Blatman and Sudret, 2010] Blatman, G. and Sudret, B. (2010). An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis. Probabilistic Engineering Mechanics, 25(2):183–197.
[Bower et al., 2009] Bower, G. C., Flanzer, T. C., and Kroo, I. M. (2009). Formation geometries and route optimization for commercial formation flight. In Proceedings of the 27th AIAA Applied Aerodynamics Conference, San Antonio, TX, USA.
[Buhmann, 2003] Buhmann, M. D. (2003). Radial Basis Functions: Theory and Implementations. Cambridge University Press.
[Camilleri, 2018] Camilleri, M. A. (2018). Aircraft operating costs and profitability. In Travel Marketing, Tourism Economics and the Airline Product, chapter 12, pages 191–204. Springer.
[Caprace et al., 2019] Caprace, D.-G., Winckelmans, G., Chatelain, P., and Eldredge, J. D. (2019). Wake vortex detection and tracking for aircraft formation flight. In Proceedings of the AIAA Aviation 2019 Forum, Dallas, TX, USA.
[Cavalier et al., 1990] Cavalier, T. M., Pardalos, P. M., and Soyster, A. L. (1990). Modeling and integer programming techniques applied to propositional calculus. Computers and Operations Research, 17(6):561–570.
[Cook et al., 2015] Cook, A., Blom, H. A., Lillo, F., Mantegna, R. N., Micciche, S., Rivas, D., Vázquez, R., and Zanin, M. (2015). Applying complexity science to air traffic management. Journal of Air Transport Management, 42:149–158.
[Dee et al., 2011] Dee, D. P., Uppala, S. M., et al. (2011). The ERA-Interim reanalysis: Configuration and performance of the data assimilation system. Quarterly Journal of the Royal Meteorological Society, 137(656):553–597.
[Durango et al., 2016] Durango, G. J., Lawson, C., and Shahneh, A. Z. (2016). Formation flight investigation for highly efficient future civil transport aircraft. The Aeronautical Journal, 120(1229):1081–1100.
[Eurocontrol, 2013] Eurocontrol (2013). User manual for the Base of Aircraft Data (BADA). Technical Report 130416, Eurocontrol.
[Fahroo and Ross, 2000] Fahroo, F. and Ross, I. M. (2000). A spectral patching method for direct trajectory optimization. Journal of the Astronautical Sciences, 48(2):269–286.
[Fahroo and Ross, 2008] Fahroo, F. and Ross, I. M. (2008). Advances in pseudospectral methods for optimal control. In Proceedings of the AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, HI, USA.
[Falck et al., 2021] Falck, R., Gray, J. S., Ponnapalli, K., and Wright, T. (2021). Dymos: A Python package for optimal control of multidisciplinary systems. Journal of Open Source Software, 6(59):2809.
[Flanzer et al., 2014] Flanzer, T. C., Bieniawski, S. R., and Blake, W. B. (2014). Operational analysis for the formation flight for aerodynamic benefit program. In Proceedings of the 52nd Aerospace Sciences Meeting, National Harbor, MD, USA.
[Flanzer et al., 2020] Flanzer, T. C., Bieniawski, S. R., and Brown, J. A. (2020). Advances in cooperative trajectories for commercial applications. In Proceedings of the AIAA SciTech 2020 Forum, Orlando, FL, USA.
[Garg, 2011] Garg, D. (2011). Advances in Global Pseudospectral Methods for Optimal Control. PhD thesis, University of Florida, Gainesville, FL, USA.
[Garg et al., 2011] Garg, D., Patterson, M. A., Francolin, C., Darby, C. L., Huntington, G. T., Hager, W. W., and Rao, A. V. (2011). Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method. Computational Optimization and Applications, 49(2):335–358.
[Garg et al., 2009] Garg, D., Patterson, M. A., Hager, W., Rao, A. V., Benson, D. A., and Huntington, G. T. (2009). An overview of three pseudospectral methods for the numerical solution of optimal control problems. Advances in the Astronautical Sciences, 135:1–17.
[Haalas et al., 2014] Haalas, D. J., Bieniawski, S. R., Whitehead, B., Flanzer, T., and Blake, W. B. (2014). Formation flight for aerodynamic benefit simulation development and validation. In Proceedings of the 52nd Aerospace Sciences Meeting, National Harbor, MD, USA.
[Hallock and Holzapfel, 2018] Hallock, J. N. and Holzäpfel, F. (2018). A review of recent wake vortex research for increasing airport capacity. Progress in Aerospace Sciences, 98:27–36.
[Hart et al., 2012] Hart, W. E., Laird, C. D., Watson, J.-P., Woodruff, D. L., Hackebeil, G. A., Nicholson, B. L., and Siirola, J. D. (2012). Pyomo-Optimization Modeling in Python. Springer.
[Hartjes et al., 2018] Hartjes, S., van Hellenberg Hubar, M. E., and Visser, H. G. (2018). Multiple-phase trajectory optimization for formation flight in civil aviation. In Dolega, B., Glebocki, R., Kordos, D., and Zugaj, M., editors, Advances in Aerospace Guidance, Navigation and Control, pages 389–405. Springer.
[Hartjes et al., 2019] Hartjes, S., Visser, H. G., and van Hellenberg Hubar, M. E. (2019). Trajectory optimization of extended formation flights for commercial aviation. Aerospace, 6(9):100.
[Héder, 2017] Héder, M. (2017). From NASA to EU: the evolution of the TRL scale in Public Sector Innovation. The Innovation Journal, 22(2):1–23.
[Hu et al., 2005] Hu, J., Prandini, M., and Sastry, S. (2005). Aircraft conflict prediction in the presence of a spatially correlated wind field. IEEE Transactions on Intelligent Transportation Systems, 6(3):326–340.
[Hull, 2007] Hull, D. G. (2007). Fundamentals of Airplane Flight Mechanics. Springer.
[Huntington, 2007] Huntington, G. T. (2007). Advancement and analysis of a Gauss pseudospectral transcription for optimal control problems. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA.
[Huntington and Rao, 2008] Huntington, G. T. and Rao, A. V. (2008). Comparison of global and local collocation methods for optimal control. Journal of Guidance, Control, and Dynamics, 31(2):432–436.
[IATA, 2013] IATA (2013). Technology roadmap, 4th Edition. Technical report, International Air Transport Association.
[IATA, 2016] IATA (2016). Press Release No. 59. IATA forecasts passenger demand to double over 20 years. https://www.iata.org/en/pressroom/pr/2016-10-18-02/. Accessed: 2021-12-19.
[ICAO, 1944] ICAO (1944). Convention on international civil aviation. Annex 2. Technical report, International Civil Aviation Organization.
[ICAO, 2016] ICAO (2016). Environmental report 2016. Aviation and climate change. Technical report, International Civil Aviation Organization.
[ICAO, 2017] ICAO (2017). Airline operating costs and productivity. Technical report, International Civil Aviation Organization.
[IPCC, 2018] IPCC (2018). Global warming of 1.5C. Technical report, The Intergovernmental Panel on Climate Change.
[Jeung et al., 2010] Jeung, H., Yiu, M. L., Zhou, X., Jensen, C. S., and Shen, H. T. (2010). Discovery of convoys in trajectory databases. Proceedings of the VLDB Endowment, 1(1):1068–1080.
[Kent and Richards, 2014] Kent, T. and Richards, A. (2014). Accounting for the effect of ground delay on commercial formation flight. In Proceedings of the 2014 UKACC International Conference on Control, Loughborough, United Kingdom.
[Kent and Richards, 2015] Kent, T. E. and Richards, A. G. (2015). Analytic approach to optimal routing for commercial formation flight. Journal of Guidance, Control, and Dynamics, 38(10):1872–1884.
[Kent and Richards, 2021] Kent, T. E. and Richards, A. G. (2021). Potential of formation flight for commercial aviation: Three case studies. Journal of Aircraft, 58(2):1–14.
[Kim and Braatz, 2012] Kim, K. K. K. and Braatz, R. D. (2012). Probabilistic analysis and control of uncertain dynamic systems: Generalized polynomial chaos expansion approaches. In Proceedings of the 2012 American Control Conference, Montreal, QC, Canada.
[Klavins, 2004] Klavins, E. (2004). Communication complexity of multi-robot systems. In Boissonnat, J., Burdick, J., Goldberg, K., and Hutchinson, S., editors, Algorithmic Foundations of Robotics V, pages 275–291. Springer.
[Kless et al., 2013] Kless, J. E., Aftosmis, M. J., Ning, S. A., and Nemec, M. (2013). Inviscid analysis of extended-formation flight. AIAA Journal, 51(7):1703–1715.
[Klower et al., 2021] Klower, M., Allen, M., Lee, D., Proud, S., Gallagher, L., and Skowron, A. (2021). Quantifying aviation’s contribution to global warming. Environmental Research Letters, 16(10):104027.
[Lai et al., 2022] Lai, Y. Y., Christley, E., Kulanovic, A., Teng, C.-C., Bjorklund, A., Nordensvard, J., Karakaya, E., and Urban, F. (2022). Analysing the opportunities and challenges for mitigating the climate impact of aviation: A narrative review. Renewable and Sustainable Energy Reviews, 156:111972.
[Lee et al., 2009] Lee, D. S., Fahey, D. W., Forster, P. M., Newton, P. J., Wit, R. C., Lim, L. L., Owen, B., and Sausen, R. (2009). Aviation and global climate change in the 21st century. Atmospheric Environment, 43(22–23):3520–3537.
[Li et al., 2014] Li, X., Nair, P. B., Zhang, Z., Gao, L., and Gao, C. (2014). Aircraft robust trajectory optimization using nonintrusive polynomial chaos. Journal of Aircraft, 51(5):1592–1603.
[Lissaman and Shollenberger, 1970] Lissaman, P. B. S. and Shollenberger, C. A. (1970). Formation flight of birds. Science, 168(3934):1003–1005.
[Liu et al., 2019] Liu, C., Jiang, B., Patton, R. J., and Zhang, K. (2019). Integrated fault-tolerant control for close formation flight. IEEE Transactions on Aerospace and Electronic Systems, 56(2):839–852.
[Marks and Gollnick, 2016] Marks, T. and Gollnick, V. (2016). Influence of aircraft type and order on fuel savings gained by two aircraft formations. In Proceedings of the 30th Congress of the International Council of the Aeronautical Science, Deajeon, South Korea.
[Matsuno et al., 2015] Matsuno, Y., Tsuchiya, T., Wei, J., Hwang, I., and Matayoshi, N. (2015). Stochastic optimal control for aircraft conflict resolution under wind uncertainty. Aerospace Science and Technology, 43:77–88.
[Michel et al., 2020] Michel, D.-T., Dolfi-Bouteyre, A., Goular, D., Augere, B., Planchat, C., Fleury, D., Lombard, L., Valla, M., and Besson, C. (2020). Onboard wake vortex localization with a coherent 1.5 μm Doppler LiDAR for aircraft in formation flight configuration. Optics Express, 28(10):14374–14385.
[Murphy, 2012] Murphy, K. P. (2012). Machine learning: A probabilistic perspective. The MIT Press.
[Ning et al., 2011] Ning, A., Flanzer, T. C., and Kroo, I. M. (2011). Aerodynamic performance of extended formation flight. Journal of Aircraft, 48(3):855–865.
[Ning, 2011] Ning, S. A. (2011). Aircraft Drag Reduction Through Extended Formation Flight. PhD thesis, Stanford University, Standord, CA, USA.
[Patterson and Rao, 2014] Patterson, M. A. and Rao, A. V. (2014). GPOPS-II: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming. ACM Transactions on Mathematical Software, 41(1):1–37.
[Prandini and Hu, 2009] Prandini, M. and Hu, J. (2009). Application of reachability analysis for stochastic hybrid systems to aircraft conflict prediction. IEEE Transactions on Automatic Control, 54(4):913–917.
[Rao et al., 2009] Rao, A. V., Benson, D., Huntington, G. T., Francolin, C., Darby, C. L., and Patterson, M. (2009). User’s Manual for GPOPS Version 2.1: A MATLAB Package for Dynamic Optimization Using the Gauss Pseudospectral Method.
[Rivas and Vazquez, 2016] Rivas, D. and Vazquez, R. (2016). Uncertainty. In Cook, A. and Rivas, D., editors, Complexity Science in Air Traffic Management. Routledge.
[Ross, 2020] Ross, I. (2020). Enhancements to the DIDO optimal control toolbox. arXiv preprint arXiv:2004.13112.
[Ross and Fahroo, 2004] Ross, I. M. and Fahroo, F. (2004). Pseudospectral knotting methods for solving optimal control problems. Journal of Guidance, Control, and Dynamics, 27(3):397–405.
[Saltelli et al., 2008] Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., and Tarantola, S. (2008). Global Sensitivity Analysis. The Primer. John Wiley & Sons.
[Seo et al., 2017] Seo, J., Kim, Y., Kim, S., and Tsourdos, A. (2017). Collision avoidance strategies for unmanned aerial vehicles in formation flight. IEEE Transactions on Aerospace and Electronic Systems, 53(6):2718–2734.
[Shapiro, 2003] Shapiro, A. (2003). Monte Carlo Sampling Methods. In Ruszczynski, A. and Shapiro, A., editors, Stochastic Programming, volume 10 of Handbooks in Operations Research and Management Science, pages 353–425. Elsevier.
[Shone et al., 2021] Shone, R., Glazebrook, K., and Zografos, K. G. (2021). Applications of stochastic modeling in air traffic management: Methods, challenges and opportunities for solving air traffic problems under uncertainty. European Journal of Operational Research, 292:1–26.
[Slotnick, 2014] Slotnick, J. P. (2014). Computational aerodynamic analysis for the formation flight for aerodynamic benefit program. In Proceedings of the 52nd Aerospace Sciences Meeting, National Harbor, MD, USA.
[Sudret, 2008] Sudret, B. (2008). Global sensitivity analysis using polynomial chaos expansions. Reliability Engineering and System Safety, 93:964–979.
[Tierno et al., 2012] Tierno, M. Á. G., Cortés, M. P., and Márquez, C. P. (2012). Mecánica del vuelo. Ibergaceta.
[Torre et al., 2019] Torre, E., Marelli, S., Embrechts, P., and Sudret, B. (2019). A general framework for data-driven uncertainty quantification under complex input dependencies using vine copulas. Probabilistic Engineering Mechanics, 55:1–16.
[Trucchia et al., 2019] Trucchia, A., Egorova, V., Pagnini, G., and Rochoux, M. (2019). On the merits of sparse surrogates for global sensitivity analysis of multi-scale nonlinear problems: Application to turbulence and fire-spotting model in wildland fire simulators. Communications in Nonlinear Science and Numerical Simulation, 73:120–145.
[Tu et al., 2008] Tu, Y., Ball, M. O., and Jank, W. S. (2008). Estimating flight departure delay distributions—a statistical approach with long-term trend and short-term pattern. Journal of the American Statistical Association, 103(481):112–125.
[Vachon et al., 2002] Vachon, M. J., Ray, R., Walsh, K., and Ennix, K. (2002). F/A-18 aircraft performance benefits measured during the autonomous formation flight project. In Proceedings of the AIAA Atmospheric Flight Mechanics Conference and Exhibit, Monterey, CA, USA.
[Vechtel et al., 2018] Vechtel, D., Fischenberg, D., and Schwithal, J. (2018). Flight dynamics simulation of formation flight for energy saving using LES-generated wake flow fields. CEAS Aeronautical Journal, 9(4):735–746.
[Vieira et al., 2009] Vieira, M. R., Bakalov, P., and Tsotras, V. J. (2009). On-line discovery of flock patterns in spatio-temporal data. In Proceedings of the 17th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, Seattle, WA, USA.
[Voskuijl, 2017] Voskuijl, M. (2017). Cruise range in formation flight. Journal of Aircraft, 54(6):2184–2191.
[Wachter and Biegler, 2006] Wachter, A. and Biegler, L. T. (2006). On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical programming, 106(1):25–57.
[Wei et al., 2008] Wei, S., Zěfran, M., and DeCarlo, R. A. (2008). Optimal control of robotic systems with logical constraints: Application to UAV path planning. In Proceedings of the 2008 IEEE International Conference on Robotics and Automation, Pasadena, CA, USA.
[Wieselsberger, 1914] Wieselsberger, C. (1914). Beitrag zur Erklärung des Winkelfluges einiger Zugvögel. Zeitschrift für Flugtechnik und Motorluftschiffahrt., 5:225–229.
[Xiu, 2010] Xiu, D. (2010). Numerical Methods for Stochastic Computations. A Spectral Method Approach. Princeton University Press.
[Xu et al., 2014] Xu, J., Ning, S. A., Bower, G., and Kroo, I. (2014). Aircraft route optimization for formation flight. Journal of Aircraft, 51(2):490–501.
[Zhu, 2019] Zhu, Y. (2019). Uncertain Optimal Control. Springer.
_______________________________
1 https://www.eurocontrol.int/covid19
2 https://www.airbus.com/en/innovation/disruptive-concepts/biomimicry/fellofly