2.1 Manual planning methods

Because of the complexities associated with SST routing, manual trajectory planning has proved to be a popular and relatively simple strategy for predicting where next-generation supersonic will operate (see Table 1). In their assessment of sonic boom-restricted flight routing, Liebhardt et al. [Reference Liebhardt, Linke and Dahlmann28] used Google Earth to manually draft routes between 250 city pairs believed most viable for the supersonic market. The work compared three scenarios for operating civil supersonic aircraft: full-permit, where supersonic flight overland is permitted and aircraft can fly great circle tracks; mixed-permit, where flight over populated areas is flown at Mach-cutoff; and no-permit, meaning that overland flight can only be flown at subsonic speeds. For the second and third scenarios, a fixed buffer of 50 km (27 nm) between the flight path and coastlines or permanent settlements was assumed on the basis that 50 km was the usual offset of former Concorde routes from coastal areas. Given that globally distributed mean atmospheric data (including winds) were used to calculate flight time and fuel burn, this fixed buffer is overly simplistic because, as already discussed, boom carpet width depends on atmospheric properties. To address this simplicity, Liebhardt et al. [Reference Liebhardt, Lütjens, Swaid, Müller and Ladewich29] subsequently used a proprietary sonic ray tracing computer code to obtain more accurate boom carpet widths. ERA-InterimFootnote ² atmospheric data for 1 January 2015, provided by the European Centre for Medium-Range Weather Forecasts (ECMWF), were used to draft a route between London and Jeddah such that the primary boom carpet did not reach the coastline (see Fig. 4). Like in Ref. (Reference Liebhardt, Linke and Dahlmann28), vertical trajectory, flight time and fuel burn were determined with a proprietary mission simulation tool. Conversely, the lateral trajectory was adjusted manually using Google Earth because at the time, full three-dimensional optimisation proved difficult. Later, Liebhardt et al. [Reference Liebhardt, Lütjens, Ueno and Ishikawa9] conducted a market assessment of JAXA’s low-boom S4 airliner, again using Google Earth to draft routes, but this time sonic boom carpets were not considered because they were assumed not to have a significant impact on the study’s outcome.

Figure 3.

Lateral propagation of shock waves from a supersonic aircraft. The downward propagating rays form the primary boom carpet whereas the rays that are bent by the upper atmosphere and subsequently impact the ground at a greater lateral distance define the secondary boom carpet. Figure obtained from Maglieri et al. [Reference Maglieri, Bobbitt, Plotkin, Shepherd, Coen and Richwine12], public use permitted.

Figure 4.

London-Jeddah manually-drafted flight path (green line) and boom carpet (white lines) for 1st January 2015. Over Egypt the aircraft was flown at Mach-cutoff, hence the discontinuity in boom carpet. Reprinted with permission from Liebhardt et al. [Reference Liebhardt, Lütjens, Swaid, Müller and Ladewich29]. Copyright 2019, DLR.

Jain et al. [Reference Jain, Ogunsina, Chao, Crossley, Ueno and DeLaurentis31] performed a similar analysis of future supersonic operations, though their focus was on 205 routes to and from the United States. Eligible routes were first identified by manually adjusting the great circle path between two airports. Incremental deviations in departure heading of $ \pm 7$ ° from the great circle generated 15 possible flight paths. Flight time for each option was calculated based on the percentage of time spent overwater and overland, along with the respective airspeeds, and the minimum-time option was taken to be the most viable. A more sophisticated routing method based on a weighted sum of time and fuel [Reference Hassan, Pfaender, Oliveira, Ougazzaden, Dussauge and Mavris34] was then used to identify the most profitable city pairs. However, it is not clear whether boom carpet width was accounted for. Mane et al. [Reference Mane, Swaine, Fung, Crossley and DeLaurentis33] assessed routes between 5317 origin-destination airport pairs using the same principle of departure heading adjustment. In common with Ref. (Reference Jain, Ogunsina, Chao, Crossley, Ueno and DeLaurentis31), the route offering the lowest flight time was assumed to be the ‘best’, and the effects of wind and time taken to accelerate and decelerate between sub- and supersonic flight were ignored.

2.2 Graph search methods

More recent works have automated lateral routing using graph search algorithms such as Dijsktra [Reference Dijkstra35], A^* [Reference Hart, Nilsson and Raphael36] and Theta^* [Reference Daniel, Nash, Koenig and Felner37] (see Table 2). This class of algorithm is typically easy to implement and scale. For flight planning applications, the graph is often a two-dimensional grid with resolution specified in terms of latitude and longitude. The optimal path from a start node to a goal node is then found for some given objective function. Penalty areas can be defined in the grid to prevent aircraft flying through restricted airspace or to stop supersonic aircraft flying overland, for example. More dimensions can be added to the graph by considering acceleration, deceleration, climb and descent, though computational effort increases with the dimensionality of the problem space.

Marsh and Spinielli [Reference Marsh and Spinielli38] created EUROCONTROL’s open-source himach tool to find minimum-time routes assuming different airspeeds overland and overwater. The tool, written in R, uses a bidirectional Dijkstra algorithm, a primitive aircraft performance model and a fixed coastal buffer to find paths that satisfy user-specified range requirements. In its current form, however, himach does not allow for non-standard atmospheric conditions or wind to be modelled. The tool is best suited to market studies and simple environmental impact modelling, and was recently used in a noise and emissions study of a proposed Mach 1.6 supersonic business jet [Reference Mourouzidis, Del Gatto, Adamidis, Villena Munoz, Lawson, Martinez Corzo, Leyland, Marsh, Lim, Owen, Terrenoire, Atinault, Legriffon, Huet, Schaefer, Plohr, Bake and Madden41].

Dijkstra’s algorithm tends to be more computationally expensive than the A^* algorithm, because with A^* a heuristic function reduces the number of grid nodes to be explored. Speth et al. [Reference Speth, Eastham, Fritz, Sanz-Morére, Agarwal, Prashanth, Allroggen and Barrett13] used an implementation of A^* to identify optimal flight routes under three scenarios: an unrestricted scenario where supersonic aircraft fly great circle tracks; a scenario where supersonic flight overland is completely prohibited; and a scenario where supersonic flight is only prohibited over areas with population density greater than 50 inhabitants per square kilometer. For the two restricted scenarios, the objective function comprised flight cost, flight time and a monetary value of time (VOT) a passenger assigns to an hour of travel time saved. Figure 5 shows the different routes produced by the algorithm for two different VOT options. In this work, only annually averaged winds from the GEOS-5Footnote ³ dataset were used for flight time and fuel burn calculations. A fixed buffer of 95 km (51 nm) around restricted areas was assumed, 95 km being the maximum carpet width for an aircraft with a maximum cruise altitude and Mach number of 19.8 km (65,000 ft) and 2.2, respectively, reported in Ref. (Reference Liebhardt, Lütjens, Swaid, Müller and Ladewich29). This buffer is arguably pessimistic given that the majority of the notional aircraft types studied were designed for lower cruise Mach numbers and altitudes. Nevertheless, the algorithm solutions in Fig. 5 highlight how changing just one parameter in an objective function can yield drastically different routes.

Figure 5.

A^* routing solutions for New York City to Mexico City assuming two different values of time. Figure obtained from Speth et al. [Reference Speth, Eastham, Fritz, Sanz-Morére, Agarwal, Prashanth, Allroggen and Barrett13], public use permitted.

Although not explicitly stated in the publications, A^* is inferred to have also been used by Freire Burgos et al. [Reference Freire Burgos, Trani, Hinze, Marien, Geiselhart, Dollyhigh and Seidel39] and subsequently Li et al. [Reference Li, Geiselhart, Palma, Patterson, Dollyhigh, Wang, Hinze and Trani40] to find minimum-time, wind-optimal routes for a low-boom SST. Atmospheric data were taken from NCEP/NCARFootnote ⁴ Reanalysis, and coastal buffers were not required due to the low-boom nature of the aircraft considered. In Ref. (Reference Li, Geiselhart, Palma, Patterson, Dollyhigh, Wang, Hinze and Trani40), three different mission profiles were simulated: a mixed mission with low-boom cruise at Mach 1.7 for overland segments and unrestricted cruise at Mach 1.8 for overwater segments; a low-boom mission with a constant cruise Mach of 1.7; and an unrestricted mission with a constant cruise Mach of 1.8. NASA’s Flight Optimization System (FLOPS)Footnote ⁵ was then used to generate a vertical flight profile. The work demonstrated how conceptual aircraft design, mission profile analysis and market demand studies can be integrated to support regulatory decision making on supersonic overland flight.

In the most recent publications, fixed buffers have been replaced in favour of buffers that vary with atmospheric and flight conditions. Efforts to integrate carpet width computation with automated boom-restricted routing were first presented by researchers at Georgia Tech. Hassan et al. [Reference Hassan, Pfaender, Oliveira, Ougazzaden, Dussauge and Mavris34] used the Theta^* algorithm to identify optimal supersonic flight routes with an objective function comprising a weighted sum of time and fuel burn. Theta^* is similar to A^*, but the addition of a line-of-sight check – in this case using Bresenham’s line of sight algorithm [Reference Bresenham42] – determines if the path between two nodes encounters an obstacle. Whereas the A^* implementation in Refs (Reference Speth, Eastham, Fritz, Sanz-Morére, Agarwal, Prashanth, Allroggen and Barrett13, Reference Freire Burgos, Trani, Hinze, Marien, Geiselhart, Dollyhigh and Seidel39) featured a 48-node neighbour scheme, Hassan et al. [Reference Hassan, Pfaender, Oliveira, Ougazzaden, Dussauge and Mavris34] used only an 8-node neighbour scheme. Without the line-of-sight check, the latter would result in tracks that can change by only $ \pm 45$ °. Theta^* therefore helps to smooth the lateral flight route while simultaneously preventing the aircraft overflying a restricted area. A baseline buffer of 27 nm was added around the edge of land masses (consistent with a 50° turn angle for Concorde), though a sensitivity study using a 150 nm buffer was also conducted. The number of routing solutions generated was found to vary with both buffer distance and airport pair. Wen et al. [Reference Wen, Pfaender and Mavris43] used PCBoom and ECMWF’s ERA5 reanalysis data for every day of 2020 to quantify the global variation of primary boom carpet width. They found that boom carpet width tends to increase for an aircraft flying (a) at higher Mach numbers and altitudes, (b) in warmer seasons regardless of flight direction, and (c) eastbound with the North Atlantic jet stream. The methods for calculating 95th percentile global carpet widths developed in Ref. (Reference Wen, Pfaender and Mavris43) were subsequently integrated with the Theta^* algorithm from Ref. (Reference Hassan, Pfaender, Oliveira, Ougazzaden, Dussauge and Mavris34) to create a bespoke flight planning tool [Reference Wen, Pfaender and Mavris44]. An artificial scaling parameter, $\alpha $ , was used to trade between time-optimal ( $\alpha = 0$ ) and fuel-optimal ( $\alpha = 1$ ) routes. The addition of acceleration/deceleration flight phases coupled with lengthy routes with complex geographic features meant that the algorithm became computationally expensive, in some cases taking up to a few hours to run. Although atmospheric data were used to estimate boom carpet width, the effect of prevailing wind does not seem to have been considered when calculating flight time or fuel burn for the different routes.

Building on previous work at DLR [Reference Liebhardt, Linke and Dahlmann28, Reference Liebhardt, Lütjens, Swaid, Müller and Ladewich29], Theta^* was also used by Liebhardt et al. [Reference Liebhardt, Ehlers and Shah45] to develop an automated routing tool. In contrast to Georgia Tech’s probabilistic approach to boom carpet estimation, the DLR toolchain allows for unique atmospheric conditions to be used as inputs. In this way, real-world operational routes that account for daily weather variations can be generated. A changeable route buffer was used to keep the supersonic aircraft sufficiently far away from the coast so as to prevent the sonic boom making landfall (see Fig. 6). Unlike the Georgia Tech tool, which allows for a trade-off between time- and fuel-optimal routes, only minimum-time routes were considered in this work. The transitions between sub- and supersonic flight were accounted for using simple averaged rates of acceleration and deceleration. While boom carpet edges were estimated using wind data from ECMWF’s ERA-Interim, it is not clear if the effect of wind on flight time was considered.

Figure 6.

Part of a route between New York City and London. A varying route buffer is used to ensure that the edge of the boom carpet does not infringe a specified coastal buffer. Reprinted with permission from Liebhardt et al. [Reference Liebhardt, Ehlers and Shah45]. Copyright 2024, DLR.

3.1 Manual planning methods

Manual planning methods have proved popular with researchers trying to understand how future supersonic aircraft will integrate with existing air traffic. Underwood and Guminsky [Reference Underwood and Guminsky30] and later Underwood [Reference Underwood46] investigated flight time savings and potential conflicts with subsonic aircraft that could arise if SSTs are permitted to fly over the contiguous United States. SST trajectories were assumed to include intermediate subsonic level-off altitudes prior to acceleration to, and after deceleration from, supersonic cruise. The level-off altitude and length were varied between FL275 and FL395, and 0 and 100 nm, respectively. Using a joint NASA-NLR tool named Traffic Manager eXecutable, 1695 different scenarios were generated and analysed. The results showed that unsurprisingly, the most conflicts occurred when SSTs were held at altitudes where there is a high density of subsonic traffic. The number of encounters was also greater during the ascent to cruise compared to the descent from cruise. On average, SST flights were found to be 53% faster than a subsonic airliner flying the same route. The study did, however, rely on the assumptions that supersonic overland flight will be permitted, and that between the end of a standard instrument departure (SID) and beginning of a standard terminal arrival (STAR), SSTs fly great circle tracks. Wind and other weather phenomena were also not simulated, despite these potentially affecting aircraft routing.

Liebhardt et al. [Reference Liebhardt, Linke and Knaack17] instead investigated how atmospheric conditions and global topography affect Mach-cutoff. Building on previous work [Reference Liebhardt, Linke and Dahlmann28, Reference Liebhardt, Lütjens, Swaid, Müller and Ladewich29], a proprietary trajectory calculation environment known as SuperTraC (supersonic trajectory calculator) was used to simulate Mach-cutoff flights between eight representative city pairs. An example for the New York City-Los Angeles route is shown in Fig. 7. Unlike in much of the literature, daily prevailing winds were considered and shown to affect the flight time of a marginally supersonic aircraft. Cruise altitude and Mach number were adjusted to ensure sonic booms did not reach the ground, and flight time savings and additional fuel burn consumption were compared to purely subsonic flight. Cutoff Mach numbers were also determined for latitudes between 60° S and 80° N, flight levels between FL330 and FL510, and for eastbound and westbound routes in January and July. However, the frequent accelerations and decelerations shown in Fig. 7 suggest that the trajectory may be suboptimal with respect to fuel burn.

Figure 7.

Mach-cutoff flight between New York City and Los Angeles with headwinds prevailing. A cutoff altitude of 5000 ft above ground level was chosen as a conservative safety buffer. Reprinted with permission from Liebhardt et al. [Reference Liebhardt, Linke and Knaack17]. Copyright 2022, DLR.

SuperTraC was used by Ishikawa et al. [Reference Ishikawa, Ueno, Koganezawa, Makino, Liebhardt and Lütjens10] to assess the under-track perceived sonic boom loudness of JAXA’s S4 low-boom SST concept. Maximum-range missions were simulated, though a zero-wind standard atmosphere was assumed, and no details of the origin or destination airports were provided. Loudness was assessed by varying between 28,000 and 42,000 ft the altitude at which the aircraft accelerates from subsonic to supersonic flight. This constitutes a novel approach given that in other works, acceleration altitude is simply specified as part of the flight profile [Reference Morgenstern, Buonanno, Yao, Murugappan, Paliath, Cheung, Malcevic, Ramakrishnan, Pastouchenko, Wood, Martens, Viars, Tersmette, Lee, Simmons, Plybon, Alonso, Palacios, Lukaczyk and Carrier47–Reference Villena Munoz, Lawson, Riaz and Jaron49]. Acceleration at higher altitude was found to give slightly quieter booms on the ground, suggesting that a combination of trajectory optimisation and innovative aircraft design will be needed to realise low-boom operations.

As well as en route sonic boom modelling, the noise generated by an SST during landing and takeoff has received considerable attention. Compared to subsonic aircraft, SSTs have higher specific thrust on account of their increased cruise Mach number and the associated increase in wave drag. The thin, highly swept wings used to minimise cruise drag typically exhibit poor low-speed flying qualities [Reference Torenbeek54], meaning that SSTs takeoff and land at faster speeds than conventional subsonic aircraft. High specific thrust and LTO speeds lead to high jet velocity and hence noise. Advanced LTO trajectories that minimise effective perceived noise level (EPNL) compared to standard trajectories are therefore of particular interest. A standard takeoff trajectory is one that conforms to the noise certification regulations for subsonic aircraft in ICAO Annex 16 Volume 1. An advanced trajectory features modification(s) to takeoff speed, automatic thrust control and/or altitude at which the aircraft’s thrust is reduced (known as the cutback altitude) [Reference Voet, Prashanth, Speth, Sabnis, Tan and Barrett55]. Nöding [Reference Nöding, Schuermann, Bertsch, Koch, Plohr, Jaron and Berton32] simulated standard and advanced takeoff procedures for an SST concept developed in Ref. (Reference Schuermann56). Airspeed, thrust and cutback height were varied, as shown in Fig. 8, and the optimum trajectory was identified as the one that provided the lowest EPNL at the flyover measurement position.

Figure 8.

A comparison between advanced (panels a, c and e) and standard (panels b, d and f) takeoff trajectories. Trajectories with the same cutback height are assigned a specific colour (grey = lower, blue = middle and black = higher cutback height). Figure obtained from Nöding et al. [Reference Nöding, Schuermann, Bertsch, Koch, Plohr, Jaron and Berton32] licenced under CC BY 4.0.

3.2 Optimal control approaches

Optimisation of supersonic aircraft vertical trajectories has been accomplished predominantly using optimal control approaches. Optimal control theory is a vast topic for which a complete overview is beyond the scope of this review. For a comprehensive introduction to optimal control problem formulation and solution, interested readers may wish to consult Refs (Reference Betts24, Reference Betts57, Reference Longuski, Guzmán and Prussing58). In general, however, an optimal control problem formulation requires definition of the following:

• • A cost functional, objective function or performance index (terms used interchangeably). The optimal control (for example, a sequence of pitch angles to fly) is designed to minimise or maximise the performance index.

• • A mathematical model of the system in the form of differential equations. These equations describe the evolution of the states of the system.

• • Boundary conditions. For example, the flight time may be ‘free’ or it may be ‘fixed’ subject to air traffic control requirements. The geographic location of origin and destination airports may also be represented as boundary conditions.

• • Constraints on the control and state variables. These can be used to limit the aircraft’s altitude (typically a state variable) and engine thrust (typically a control variable), for example.

Optimal control problems generally do not have closed-form analytic solutions and are therefore solved numerically using indirect (‘optimise then discretise’) or direct (‘discretise then optimise’) methods. Indirect methods are so named because they solve a different optimisation problem (a boundary value problem) that indirectly represents the original optimal control problem. Indirect methods are based on the calculus of variations and require derivation of the necessary conditions for optimality. This, as well as the need to make initial guesses for the adjoint variables, can prove difficult. On the other hand, direct methods discretise the problem directly and convert it into a finite-dimensional non-linear programming (NLP) problem that can be solved using efficient non-linear optimisation algorithms. Direct methods are regarded as easier to set up and solve than indirect methods, but they lack the in-built accuracy metric of an indirect method [Reference Betts57].

It should be noted that the majority of early works optimised trajectories for military supersonic fighter-interceptor aircraft. A seminal paper by Bryson et al. [Reference Bryson, Desai and Hoffman59] demonstrated how the energy-state approximation could be applied to the minimum time-to-climb and minimum fuel-to-climb problems but then also extended this to maximum-range problems. The energy-state approximation simplifies such problems by using only one state variable: energy per unit mass, or specific energy. Importantly, the work highlighted for the first time the counterintuitive nature of optimal supersonic trajectories in that such an aircraft must actually dive first before climbing to a given altitude in minimal time. Despite its relative simplicity, the energy-state approximation was shown to give similar optimal climb profiles to more complex (up to five-state) models [Reference Shaw60]. However, others have argued that the energy-state approximation is best suited to transport aircraft due to their benign manoeuvres making the assumptions involved in the approximation ‘less questionable’ [Reference Ardema, Windhorst and Phillips61]. Nevertheless, Bryson et al.’s work [Reference Bryson, Desai and Hoffman59] led to significant interest in the optimisation of high-performance fighter aircraft trajectories. Subsequent research found two-dimensional minimum-time turns [Reference Hedrick and Bryson62], three-dimensional minimum-time [Reference Hedrick and Bryson63, Reference Lin64] and minimum-fuel turns [Reference Hedrick and Bryson65], and maximum-range trajectories [Reference Seywald, Cliff and Well66]. More recently, Nevrekar et al. [Reference Nevrekar, Striz and Vedula67] and Levin and DeLaurentis [Reference Levin and DeLaurentis68] maximised glide range, and Jasa et al. [Reference Jasa, Brelje, Gray, Mader and Martins69] and Jasa et al. [Reference Jasa, Mader and Martins70] optimised entire flight trajectories subject to thermal management system constraints. The minimum time-to-climb problem has also remained popular, with contemporary textbook [Reference Betts57] and MATLAB [Reference Nie, Faqir and Kerrigan71] examples providing modern, accessible solutions.

3.2.2 Direct optimal control methods

In more recent works, direct methods have been employed to optimise takeoff and climb profiles of SST aircraft. In their determination of an aircraft’s optimum cruise altitude, Kang and Ryu [Reference Kang and Ryu82] used the Hermite-Simpson direct collocation method followed by the interior point solver, IPOPT,Footnote ⁶ to solve the resulting NLP problem. The minimum time-to-climb and minimum fuel-to-climb trajectories were obtained, and the altitude at which the difference between these two trajectories exceeded an acceptable error level was taken to be the multi-objective cruise altitude (MOCA). Confusingly, however, performance data for a supersonic aircraft were only used due to a lack of subsonic aircraft data. The MOCA of 10.91 km (approximately 36,000 ft) quoted may be appropriate for a subsonic airliner but is significantly lower than the cruise altitude expected for future supersonic transports. Although trajectories for two supersonic aircraft were presented for altitudes up to 20 km (approximately 66,000 ft, see Fig. 9), flight path and Mach number trends above MOCA were not discussed, even though they displayed similar transonic dives to those seen in previous work [Reference Windhorst, Ardema and Kinney79, Reference Windhorst and Ardema81].

Figure 9.

Minimum-time and minimum-fuel climbs. A multi-objective cruise altitude was selected based on the altitude at which the two different trajectory types diverge. Figure obtained from Kang and Ryu [Reference Kang and Ryu82] licenced under CC BY 4.0.

Pseudospectral collocation methods [Reference Garg, Patterson, Hager, Rao, Benson and Huntington83] have increased in popularity in the past two decades, especially in the aerospace sector, and have been used to optimise the takeoff trajectories of SSTs. Voet [Reference Voet84] investigated the extent to which variable noise reduction systems (VNRS) could minimise EPNL during takeoff. A Gauss-Lobatto pseudospectral collocation method implemented in NASA’s DymosFootnote ⁷ was used to discretise the optimal trajectory problem, and IPOPT was subsequently used to solve the NLP problem. VNRS trajectories with automatic control of thrust and high-lift devices were shown to offer reductions in EPNL, but the most effective measure was found to be an increased takeoff speed. Although runway length was assumed not to be a limiting constraint, takeoff speeds on the order of 250 knots would likely limit airports from which SSTs can operate. Horton and Busch [Reference Horton and Busch85] compared VNRS trajectories generated using NASA’s new open-source design framework Aviary, which incorporates Dymos trajectory components, with trajectories generated using the legacy FLOPS tool. In this work, a Radau pseudospectral method and Sparse Nonlinear OPTimizer (SNOPT,Footnote ⁸ a sequential programming approach) were used. Aviary VNRS trajectories provided additional takeoff EPNL reductions of between 0.8 and 1.3 EPNdB, though concerns over computational speed and robustness of the new tool warrant further development.

Despite the renewed interest in supersonic flight and sonic boom computation, Falco’s 1963 paper [Reference Falco86] remains one of the few examples of an optimal control approach to supersonic trajectory optimisation that includes a constraint on sonic boom intensity. Minimum-fuel acceleration climb paths were obtained using a gradient penalty function technique [Reference Kelley87] with a simplified overpressure formula [Reference Carlson88] added as an inequality constraint. Compared to a trajectory with no regard for overpressure limitations, the climb trajectory optimised for an overpressure limit of 2.25 pounds per square foot burned almost twice as much fuel. This corresponded to a net range reduction of ‘approximately 500 statute miles’ (434 nm), highlighting the commercial difficulties arising from overland supersonic flight restrictions.

3.3 Metaheuristic methods

To date, there has been limited application of metaheuristic approaches to SST vertical trajectory optimisation. Research has focused on the use of genetic algorithms to optimise landing trajectories. Inspired by biological evolution, genetic algorithms assume that a trajectory is characterised by discrete variables, and for a finite number of samples, randomly select values for the unknown problem variables. The ‘optimum’ trajectory is that which provides the best solution after all samples have been analysed. Genetic algorithms are typically straightforward to implement, but their randomness means that convergence to a globally optimal solution is not guaranteed [Reference Betts24, Reference Dancila and Botez89].

Kanazaki and Saisyo [Reference Kanazaki and Saisyo50] optimised the flight path of a delta-winged SST from a height of 500 m (1640 ft) down to the surface. Uniquely, the cost function comprised distance, control surface deflection and flight path angle. The algorithm employed in the study used a real-coded representation, simulated binary crossover and polynomial mutation. The work was developed further in Kanazaki et al. [Reference Kanazaki, Setoguchi and Saisyo52] with the introduction of additional controls and consideration of wind. Two cases were considered: landing using elevator control only, and landing using elevator and thrust controls. Both still air and microburst conditions were analysed, the latter relying on a simple mathematical model. The Constrained Non-dominated Sorting Genetic Algorithm II (CNSGA-II) was used to solve each problem with a population size of 30 and 30,000 generations. Compared to the still air condition, the differences in the controls among the objective functions were lower under microburst conditions, from which it was suggested that pilots’ controls should be limited during unexpected weather. The operational implications of this were not discussed. In addition, the work is somewhat limited in that it focuses on a very short time period (flight from 250 m, or 820 ft, down to the surface) and models only one microburst position (2 km, or 1.1 nm, from the aircraft’s starting point).

Takubo and Kanazaki [Reference Takubo and Kanazaki53] noted that the problem transcription used in Ref. (Reference Kanazaki, Setoguchi and Saisyo52) could lead to overly aggressive flight control design under meteorological uncertainties. To address this, non-intrusive polynomial chaos expansion was introduced to formulate a robust constrained multi-objective optimal control problem which was subsequently solved using CNSGA-II, with polynomial mutation and simulated binary crossover for the evolution strategies. Uncertainty in wind velocity was modelled over the interval [-5,5] m/s. Compared to nominal trajectories, robust trajectories were shown to minimise the aircraft’s tendency to touchdown ahead of the runway (i.e. crash) in the event of a headwind. The wind velocities modelled, however, are low compared to what might be expected in a real operational environment.

4.1 Aircraft design environments

A variety of tools have been used to estimate fuel burn for clean-sheet supersonic aircraft designs (see Table 4). Here, an overview of the software developed by different organisations is presented, and some of the limitations are discussed.

Table 4.

Software tools used to estimate fuel burn for supersonic aircraft designs

Researchers at Cranfield University used the GENUS multidisciplinary design analysis and optimisation environment to develop a number of conceptual supersonic aircraft designs [Reference Sun and Smith48, Reference Sun and Smith98, Reference Sun and Smith99]. The propulsion module of GENUS uses the NASA EngineSim model to build a rubber engine system. Given user-specified engine geometry, bypass ratio, materials and temperature limits, EngineSim provides specific fuel consumption (SFC) as a function of Mach number and altitude, from which total fuel burn can be calculated. Compared to other supersonic concepts found in the literature, however, the cruise SFC of 47.74 g/kN/s given in Sun and Smith [Reference Sun and Smith48] appears quite high. A fuel consumption per seat-km flown of up to 11.6 times the subsonic baseline is also greater than found in other sources, with Rutherford et al. [Reference Rutherford, Eastham, Sanz-Morère, Kim and Speth97] instead estimating only a seven to nine times increase, for example.

Windhorst et al. [Reference Windhorst, Ardema and Kinney79] developed a new trajectory optimiser for a similar design environment, NASA’s AirCraft SYNThesis (ACSYNT) code. ACSYNT includes a subroutine that generates a trajectory for each mission segment, then simulates a vehicle flying these segments by integrating the vehicle’s equations of motion. In this work, trajectories between London and New York were analysed for a modified Boeing High Speed Civil Transport. Although some aircraft parameters were specified, fuel flow rate was simply stated to be a non-linear function of throttle, altitude and Mach number, but the function itself was not provided.

Georgia Tech’s Framework for Advanced Supersonic Transport (FASST) was used in [Reference Hassan, Pfaender, Oliveira, Ougazzaden, Dussauge and Mavris34] to derive average sub- and supersonic specific air range (SAR) cruise values for a 55-seat Mach 2.2 SST. Although SAR is a function of instantaneous weight and altitude, average values were used for simplicity. In common with other aircraft design and simulation software, FASST combines multiple modules. For example, engine cycle modelling is performed by NASA’s Numerical Propulsion System Simulation (NPSS), and mission analysis is conducted using NASA’s FLOPS [Reference Baltman, Tai, Shi and Mavris100]. Access to both NPSS and FLOPS is, however, controlled by NASA.

DLR’s flight performance code SuperTraC simulates subsonic-supersonic missions by iteratively solving the equations of motion [Reference Liebhardt, Linke and Dahlmann28, Reference Liebhardt, Ehlers and Shah45]. Details about the tool are scarce, but it is said to require aircraft performance specifications including masses, aerodynamic polars, engine maps and segment speeds to determine the flight trajectory [Reference Liebhardt, Lütjens, Swaid, Müller and Ladewich29]. Like GENUS, ACSYNT and FASST, SuperTraC is proprietary, and published works that have used the tool have all relied on aircraft performance data that are not in the public domain [Reference Ishikawa, Ueno, Koganezawa, Makino, Liebhardt and Lütjens10, Reference Liebhardt, Linke and Knaack17, Reference Liebhardt, Ehlers and Shah45].

Researchers at the University of Michigan instead demonstrated how open-source tools can simultaneously optimise a supersonic aircraft’s design and flight path. Jasa et al. [Reference Jasa, Brelje, Gray, Mader and Martins69] used OpenMDAO as the underlying optimisation framework, as well as a variety of other tools written in Python, to minimise fuel burn subject to thermal constraints. Their propulsion module, for example, uses the open-source tool pyCycle which has been verified against NPSS [Reference Hendricks and Gray101]. Although this work focused on optimising the trajectory for a reconnaissance flight (i.e. a military application), it was suggested that a similar method could be used to assess a wide variety of mission types.

4.2 Concorde performance data

In other works, fuel burn has been estimated using historic Concorde data. Gonzalez-Gallego [Reference Gonzalez-Gallego, Perez and Jansen102] calibrated the turbojet SFC model from Ref. (Reference Mattingly, Heiser, Boyer, Haven and Pratt103) using data for the Rolls Royce-Snecma Olympus 593 (Concorde’s engine). Aerodynamic and propulsive efficiency factors were then varied to quantify the technology improvements required for a feasible supersonic business jet. A similar calibration approach was taken by Hassan et al. [Reference Hassan, Pfaender and Mavris104] when creating their conceptual mission analysis tool, but in this work both turbojet and turbofan engine models were considered. The fuel required for different flight phases was estimated using the weight fraction method from Ref. (Reference Mattingly, Heiser, Boyer, Haven and Pratt103), and takeoff weight reductions relative to Concorde were calculated for assumed improvements in SFC and drag coefficient. The tool was subsequently used to estimate acceleration, takeoff and climb, and descent and landing fuel burn for the Georgia Tech 55-seat Mach 2.2 SST described in Section 4.1 [Reference Hassan, Pfaender, Oliveira, Ougazzaden, Dussauge and Mavris34].

Researchers undertaking market assessments and environmental impact evaluations as part of FAA ASCENT Project 010 [Reference Mavris, Crossley, Jimmy and DeLaurentis105] have tended to use simple performance models based on data from the Concorde flight manual. Guided by information about Concorde, Jain et al. [Reference Jain, Ogunsina, Chao, Crossley, Ueno and DeLaurentis31] used ‘engineering judgement’ to estimate the supersonic SFC of the aforementioned 55-seat Mach 2.2 SST. A corresponding subsonic SFC was calculated by multiplying together the supersonic SFC and Concorde’s subsonic-to-supersonic SFC ratio. A similar albeit slightly more sophisticated method was employed by Weit et al. [Reference Weit, Wen, Anand, Mayakonda, Zaida and Mavris106] and Wen et al. [Reference Wen, Weit, Anand, Mayakonda, Zaidi and Mavris51]. In these works, specific distance was presented as a function of flight condition (sub- or supersonic) and mass, and acceleration fuel burn was given simply as a function of mass. Both specific distance and acceleration fuel burn values were scaled from the Concorde flight manual. The scaling factor was determined using high-fidelity mission analysis of the 55-seat Mach 2.2 SST, and the simple fuel burn method was found to give accurate enough estimates for a first-order analysis. Supersonic-to-subsonic fuel flow rate ratios from the flight manual were then used to estimate the NOx emissions indices during subsonic cruise and transonic acceleration in accordance with the Boeing Fuel Flow Method 2 [Reference DuBois and Paynter107].

4.3 Look-up tables

Aircraft performance look-up tables are a fast and convenient way to estimate fuel burn. BADA’s performance summaries, for example, provide fuel flow rate for different flight phases, altitudes and airspeeds. The BADA generic military fighter model (denoted FGTN) was used in Ref. (Reference Underwood and Guminsky30) to investigate commercial supersonic transport operations in North American airspace. However, according to the BADA user manual [Reference Nuic108], the FGTN model is representative of a variety of small sub- and supersonic fighter aircraft. Using the FGTN model as a surrogate for a future supersonic transport is, as Underwood and Guminsky [Reference Underwood and Guminsky30] admit, a major limitation of the study, though they appear to have only taken airspeed and climb rate data from BADA. BADA’s use is regulated through a licence agreement with EUROCONTROL which, coupled with the current lack of representative SST model, makes it unappealing for supersonic trajectory optimisation research.

Look-up tables also feature in older trajectory optimisation works, but the origins of the data are often obscure or poorly explained. Wilson [Reference Wilson75] relied on Concorde aerodynamic and engine information provided by the British Aircraft Corporation, but these data were not published due to their proprietary nature. Teren and Daniele [Reference Teren and Daniele77] presented a plot of SFC as a function of thrust coefficient for an engine operating with and without afterburner. Their aircraft engine data were ‘representative of a supersonic transport’, though it is unclear where these data originated. Falco [Reference Falco86] used the fact that fuel flow is a function of Mach number and altitude to obtain minimum-fuel acceleration-climb profiles for an SST but provided no details of the function itself. Bryson et al. [Reference Bryson, Desai and Hoffman59], on the other hand, included look-up tables for thrust and fuel consumption as a function of Mach number and altitude for two supersonic interceptor aircraft. These aircraft were subsequently used in Ref. (Reference Kang and Ryu82) as proxies for commercial aircraft; however, this was due to a lack of accessible contemporary data. Baber [Reference Baber109] provided a comprehensive set of aerodynamic and performance data for a transpacific, Mach 2.6, 273-seat supersonic concept aircraft, but the report is relatively old (published in 1979) and the aircraft’s design mission is substantially different to those proposed in contemporary literature [Reference Mourouzidis, Del Gatto, Adamidis, Villena Munoz, Lawson, Martinez Corzo, Leyland, Marsh, Lim, Owen, Terrenoire, Atinault, Legriffon, Huet, Schaefer, Plohr, Bake and Madden41, Reference Sun and Smith48, Reference Villena Munoz, Lawson, Riaz and Jaron49, Reference Li and Geiselhart110].

4.4 Non-proprietary performance data

Rather than relying on proprietary models or Concorde data, NASA designed a 55-tonne supersonic technology concept aeroplane (STCA) using information entirely from the public domain [Reference Berton, Huff, Geiselhart and Seidel111, Reference Berton112]. Developed primarily for ICAO environmental and noise studies, the STCA features three CFM56-derived low-bypass ratio turbofan engines. Noise and performance data for the aircraft were shared with researchers undertaking takeoff trajectory optimisation work [Reference Voet84, Reference Voet, Prashanth, Speth, Sabnis, Tan and Barrett113]. The STCA is, however, relatively small (eight passengers) and slow (Mach 1.4 cruise) compared to other civil supersonic concepts found in the literature. Speth et al. [Reference Speth, Eastham, Fritz, Sanz-Morére, Agarwal, Prashanth, Allroggen and Barrett13] used the STCA and an updated version of Concorde [Reference Roxburgh114] as reference aircraft, and from these defined eight notional supersonic aircraft. These notional aircraft have cruise Mach numbers between 1.4 and 2.2 and capacity for 20–100 passengers. Cruise SFC, lift-to-drag ratio ( $L/D$ ) and operating empty weight were specified for each notional aircraft in line with past supersonic aircraft concepts (see Fig. 10). However, very few concepts of similar design mission from the literature were considered; notable omissions are the aircraft developed as part of the Environmentally Friendly High Speed Aircraft (HISAC) project for which performance characteristics are published [Reference De Saint Martin115]. To calculate fuel burn, the flight profile developed by Morgenstern et al. [Reference Morgenstern, Buonanno, Yao, Murugappan, Paliath, Cheung, Malcevic, Ramakrishnan, Pastouchenko, Wood, Martens, Viars, Tersmette, Lee, Simmons, Plybon, Alonso, Palacios, Lukaczyk and Carrier47] in response to NASA’s ‘N+2 Supersonic Validation: Advanced Concept Studies for Supersonic Commercial Transports Entering Service in the 2018–2020 Period’ contract was assumed. SFC and $L/D$ values for phases of flight other than cruise were scaled using ratios from Refs (Reference Morgenstern, Buonanno, Yao, Murugappan, Paliath, Cheung, Malcevic, Ramakrishnan, Pastouchenko, Wood, Martens, Viars, Tersmette, Lee, Simmons, Plybon, Alonso, Palacios, Lukaczyk and Carrier47, Reference Berton, Huff, Geiselhart and Seidel111).

Figure 10.

Performance characteristics for six notional supersonic aircraft, compared to existing designs. Figure obtained from Speth et al. [Reference Speth, Eastham, Fritz, Sanz-Morére, Agarwal, Prashanth, Allroggen and Barrett13], public use permitted.

Janić [Reference Janić116] adopted an even simpler approach to fuel burn calculation using aerodynamic data for the Reaction Engines Mach 5 A2 concept that was developed as part of the EU-funded Long-Term Advanced Propulsion Concepts and Technologies (LAPCAT) programme. A basic climb-cruise-descent trajectory was assumed, and the average thrust for each of the three flight phases was determined analytically. Multiplying the thrust by the length of each segment gave the energy requirement and, by accounting for propulsive efficiency and fuel specific energy, the fuel consumption. This work focused on network-scale infrastructural, technical, operational, economic, environmental, and social impacts of supersonic aircraft, so a high-fidelity fuel burn model was arguably not required. However, it would have been straightforward to split the flight profile into smaller segments to avoid averaging values over large ( $ \gt $ 6000 nm) distances.