Framework for Initializing the Conceptual Design of Hypersonic Aircraft

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Framework for Initializing the Conceptual Design of Hypersonic
1
Aircraft2
Daniel M. Wilson3
Liberty University, Lynchburg, Virginia 245154
Richard Figliola5
Clemson University, Clemson, South Carolina 296346
Wayne Strasser7
Liberty University, Lynchburg, Virginia 245158
José Camberos9
U.S. Air Force Research Laboratory, Dayton, OH 4543310
To assist in initializing the conceptual design of hypersonic aircraft, we outline a new,
11
systematic framework based on historical aircraft data and primarily composed of design
12
data and regression models. It is a rapid, low-fidelity analysis to provide a starting point for
13
the conceptual design process by 1) assessing the performance capabilities of four types of
14
high-speed aircraft, 2) providing initial estimates for weights and geometry with uncertainty,
15
and 3) exploring how changes in these affect performance within design spaces. Using this
16
framework, an initial set of reasonable aircraft configurations is obtained based on speed,
17
altitude, and payload requirements, which can serve to accelerate the design process and avoid
18
unforeseen problems later in the design cycle. An example is provided to demonstrate the
19
application of the framework to launch the conceptual design of a new hypersonic aircraft with
20
a given set of mission requirements.21
Keywords: aerospace, high-speed, design space, regression22
Nomenclature23
𝐴𝑚𝑎 𝑥 = maximum altitude, m
𝐴𝑅 = aspect ratio; 𝑏2/𝑆𝑝𝑙𝑎𝑛
𝑏= wingspan, m
𝑓 𝑓 = fuel fraction; 𝑊𝐹/𝐺𝑇 𝑂𝑊
𝐺𝑇 𝑂𝑊 = gross takeoff weight, N
𝐾𝑤= wetted-to-planform area ratio; 𝑆𝑤𝑒𝑡 /𝑆𝑝𝑙 𝑎𝑛

𝐿= length, m
𝑀𝑚𝑎 𝑥 = maximum Mach number
𝑂𝑊 𝐸 = operating weight empty, N
𝑝 𝑓 = payload fraction; 𝑊𝑃/𝐺𝑇𝑂𝑊
𝑠 𝑓 = structural factor; 𝑊𝐸/𝐺𝑇 𝑂𝑊
𝑆𝑝𝑙𝑎𝑛 = planform area (primary lifting surface), m2
𝑆𝑤𝑒𝑡 = wetted area, m2
𝜏= Küchemann 𝜏parameter; 𝑉𝑡𝑜𝑡 𝑎𝑙 /𝑆1.5
𝑝𝑙𝑎𝑛
𝑢= uncertainty of estimate (indicated by subscript, units vary)
𝑉𝑡𝑜𝑡 𝑎𝑙 = total volume, m3
𝑊𝐸= empty weight, N
𝑊𝐹= fuel weight, N
𝑊𝑃= payload weight, N
I. Introduction24
Since
Wilbur and Orville Wright first flew their airplane across the sands near Kitty Hawk [
1
], humanity has
25
continually sought to push the bounds of flight. Advances in aircraft technology and progress in fundamental
26
research have enabled the development of aircraft that fly progressively faster and higher, leading to one of the most
27
important frontiers of flight: hypersonics. While hypersonic aircraft have been designed and flown for decades, most
28
hypersonic aircraft have been experimental, leaving practical and commonplace hypersonic flight as a goal for the future.
29
Such technology has countless applications, including space access and global transportation.30
Hypersonic speeds are generally considered greater than Mach 5, although this is not a strict definition [
2
].
31
Rather, hypersonic flow occurs in the flight regime where a number of phenomena become important. Four major
32
characteristics of hypersonic flow over a body are described in [
2
]: a thin shock layer, an entropy layer, viscous
33
interaction, and high-temperature flows. Additionally, hypersonic aircraft can experience low-density flow when flying
34
at high altitudes, although this is not a result of their speed. For the aircraft designer, difficulties arise from the
35
aforementioned characteristics of hypersonic flow, driving a need for substantially different configurations for hypersonic
36
vehicles compared to subsonic and supersonic vehicles [2].37
The most significant aspect of hypersonic aircraft designs, distinguishing them from subsonic and supersonic designs,
38
is the integrated nature of various aircraft components [
2
]. From a general survey of aircraft, lift, propulsion, and
39
volume components tend to be distinct for subsonic and supersonic aircraft, whereas for hypersonic aircraft, these three
40
components tend to be closely integrated in an overall lifting shape [
2
]. Another aspect of hypersonic aircraft design is
41
2

the different parameters that tend to become more significant during the design process. Volume becomes increasingly
42
important with higher speeds, as this has significant implications for drag [
3
,
4
]. A greater volume of fuel could be
43
necessary to reach higher speeds, but this must be balanced with the need to reduce drag. As a result, the Küchemann
𝜏44
parameter (
𝜏=𝑉𝑡𝑜𝑡 𝑎𝑙 /𝑆1.5
𝑝𝑙𝑎𝑛
) is considered an important hypersonic parameter [
5
], a sort of “volume loading” along the
45
lines of the traditional wing loading parameter (
𝐺𝑇 𝑂𝑊 /𝑆𝑝𝑙𝑎𝑛
). Here,
𝑉𝑡𝑜𝑡 𝑎𝑙
is total volume,
𝑆𝑝𝑙𝑎𝑛
is planform area,
46
and
𝐺𝑇 𝑂𝑊
is gross takeoff weight.
𝜏
is also a slenderness parameter, as a larger value results in a bulkier aircraft, and a
47
smaller value results in a more slender aircraft [
3
].
𝜏=0.1
is considered reasonable for hypersonic cruise vehicles
48
using liquid fuels and oxidizers [3]. The two extremes for 𝜏would be 0 for a flat sheet and 0.752 for a sphere [4].49
The aircraft design process, often a long and rigorous process with many design iterations, begins with mission
50
requirements, in particular, how the aircraft is desired to perform based on its purpose. Broadly, the aircraft design
51
process can be divided into three parts: conceptual design, preliminary design, and detailed design. Through each phase,
52
the detail of the design is increased, and all three phases are driven by mission requirements. There is a plethora of
53
software to aid in conceptual design, including, but not limited to, software for analyzing weights, trajectory, propulsion,
54
controls, and aerodynamics. However, it can be difficult to begin conceptual design of a hypersonic vehicle without any
55
sense of what a reasonable design might entail.56
We outline a means to establish a starting point from which a more complete concept can be formulated. This
57
process is referred to here as “conceptual design initialization.” It is a low-fidelity initial concept assessment that bridges
58
the gap between a design proposal (with mission requirements) and the conceptual design process, as illustrated in Fig.
59
1. The designer can get an initial feel for reasonable weight and geometry estimates, putting numbers on a page and
60
beginning initial sketches for designs. From this starting point, the designer can proceed through the conceptual design
61
process, using higher fidelity models to further refine parameter estimates and assess other aircraft characteristics. Such
62
an initialization process can serve to accelerate the design process, avoid unforeseen problems later in the design cycle,
63
provide initial parameter estimates that can later be refined, and allow initial design space exploration. To clarify, “design
64
space exploration” in this paper does not refer to the exploration of space as in the Apollo missions. Rather, it refers to
65
the aircraft design process: feeling out reasonable combinations of design parameters (design spaces). Significantly
66
altering designs late in the design cycle results in serious financial costs and lost time, but a reasonable starting point
67
can mitigate this risk. Current approaches to hypersonic design initialization rely on interpolation between existing
68
designs or simply the experience of the designer. Neither of these is particularly methodical. There is a need for clear,69
quantitative, and practical initialization methods for hypersonics.70
The initialization approach taken here uses information about past hypersonic aircraft to inform new designs and
71
does so by modeling historical aircraft data. This is essentially a machine learning approach: collecting information on
72
past design decisions, detecting patterns, and using predictive models to make future design decisions for new aircraft.
73
Several supervised machine learning algorithms are employed to this end. The framework thus has a basis in real
74
3

Fig. 1 The aircraft design process with conceptual design initialization.
hypersonic aircraft that have flown, along with the extensive experience and testing to produce them. Additionally, the
75
use of regression models provides a means of capturing uncertainty, providing a confidence level for initial estimates.
76
The primary limitation of this approach for hypersonic aircraft design is the lack of historical data. Few aircraft have
77
flown at hypersonic speeds, which results in few points of reference. To mitigate, some hypersonic aircraft were included
78
in our database that did not fly but reached detailed levels of design.79
This approach is loosely modeled after an Air Force Research Laboratory report [
6
] that provides design data and
80
procedures for beginning the conceptual design of subsonic and supersonic military aircraft by establishing an initial
81
aircraft configuration. These procedures are largely based on an extensive collection of historical aircraft data and
82
primarily expressed with simple, single variable regression models [
6
]. However, no such methods (i.e. using historical
83
aircraft data) exist for hypersonics, which is a gap in current hypersonic design procedures. Our work outlines a novel
84
initialization process for hypersonics with several unique inclusions: 1) the application of machine learning algorithms,
85
2) uncertainty quantification for weight and geometry estimates, and 3) design space exploration. Beyond single variable
86
regression models, machine learning algorithms were needed for both classification and and multivariable regression.
87
While some of the analysis methods and regression model techniques are informed by the plethora of aircraft design
88
resources for subsonic and supersonic aircraft, all models are uniquely constructed with hypersonic aircraft data, and the
89
framework as a whole is uniquely designed for hypersonics. The framework, data, and models presented are a new,
90
relevant, and applicable contribution to hypersonic aircraft design literature.91
This initialization framework is outlined specifically for rocket-powered, air-launched hypersonic vehicles but can
92
theoretically be extended to other types of hypersonic vehicles. While historical data were collected for four types of
93
hypersonic vehicles, data for rocket-powered, air-launched vehicles was found to be the most extensive and useful for
94
modeling. All regression models presented here are for rocket-powered, air-launched hypersonic vehicles (although a
95
4

classification model is presented for four types of hypersonic vehicles). The majority of aircraft data for all figures in
96
this paper was obtained from
[14]
as these aircraft were often experimental in nature. The time range for this historical
97
data is from the 1950s until the present.98
The initialization framework is divided into three parts, summarized in Fig. 2. Part 1, vehicle type assessment, is
99
to determine which vehicle type is most suitable for achieving desired performance metrics. This step can be used
100
to confirm (or question) that an already selected vehicle type is suitable or to select an appropriate vehicle type for a
101
mission, realizing that different types of hypersonic vehicles have significantly different capabilities. Part 2, postulating
102
an initial reference configuration, is to obtain a set of initial, low-fidelity estimates for geometry (size) and weights that
103
includes uncertainty bounds. These estimates comprise an “initial reference configuration" for the aircraft design. Part
104
3, initial design space exploration, is to observe how design changes affect performance by constructing design spaces
105
with select parameters. Recognizing that multiple designs could fulfill a given set of mission requirements, this step is
106
to identify those other design possibilities and the effects of varying designs.107
Fig. 2 Three parts of the conceptual design initialization framework for hypersonic aircraft.
II. Methodology108
General aircraft design begins with mission requirements and seeks to provide output towards a design that meets
109
those requirements. For the purposes of this initialization framework, performance characteristics refer to those
110
parameters that describe how an aircraft performs (synonymous with mission requirements), and design parameters refer
111
to those parameters that describe how an aircraft is constructed (here weights and geometry). Thus, for initialization,
112
performance characteristics are the inputs, and design parameters are the outputs. The initialization framework uses
113
three performance characteristics for the mission requirements and estimates twelve design parameters: performance
114
characteristics =
𝑀𝑚𝑎 𝑥
,
𝐴𝑚𝑎 𝑥
,
𝑊𝑃
, and design parameters =
𝐺𝑇 𝑂𝑊
,
𝑊𝐸
,
𝑊𝐹
,
𝑠 𝑓
,
𝑓 𝑓
,
𝐿
,
𝑏
,
𝑆𝑝𝑙𝑎𝑛
,
𝐴𝑅
,
𝑉𝑡𝑜𝑡 𝑎𝑙
,
𝑆𝑤𝑒𝑡
,
115
𝐾𝑤
. Here,
𝑀𝑚𝑎 𝑥
is maximum Mach number,
𝐴𝑚𝑎 𝑥
is maximum altitude, and
𝑊𝑃
is payload weight.
𝑊𝐸
is empty
116
weight,
𝑊𝐹
is fuel weight,
𝑠 𝑓
is structural factor,
𝑓 𝑓
is fuel fraction,
𝐿
is length,
𝑏
is wingspan,
𝐴𝑅
is aspect ratio,
117
𝑆𝑤𝑒𝑡
is wetted area, and
𝐾𝑤
is wetted-to-planform area ratio. Planform area is used here to mean the area of the primary
118
lifting surface of the aircraft. For most aircraft, the wings alone serve this purpose, but for hypersonic aircraft, significant
119
lift is often generated under the entire bottom surface of the aircraft because of higher pressure under the body generated
120
5

by shock wave compression [2].121
A. Vehicle Type Assessment122
Data for four distinct hypersonic vehicle types are compared. These are primarily distinguished based on propulsion
123
method and launch method, labeled by “propulsion method/launching method." They are 1) Rocket/Air-Launched, 2)
124
Air-Breather/HTO (Horizontal Takeoff), 3) Air-Breather/Air-Launched/Booster, and 4) Glider/Rocket-Boosted. These
125
categories were chosen based on a general survey of historical high-speed aircraft in an effort to meaningfully distinguish
126
data for regression models.127
A “Rocket/Air-Launched" hypersonic vehicle is carried up to altitude under a larger, slower aircraft (like a bomber)
128
and then released to fly using rocket propulsion. For the purposes of this analysis, the fuel weight for this vehicle type
129
includes both fuel and oxidizer (equivalent to propellant) in order to generalize the parameter
𝑊𝐹
across all vehicle
130
types. An “Air-Breather/HTO" hypersonic vehicle takes off horizontally on a runway and is powered by an air-breathing
131
engine. Most modern aircraft fall into this category, and while these include some high-speed aircraft, none have flown
132
at Mach 5. An “Air-Breather/Air-Launched/Booster" hypersonic vehicle is attached to the head of a rocket, which is
133
carried up to altitude under a larger, slower aircraft. The aircraft and rocket booster is then released, at which time the
134
rocket boosts the aircraft up to an even higher speed. Finally, the aircraft separates from the rocket booster and flies
135
using scramjet propulsion. A “Glider/Rocket-Boosted" hypersonic vehicle (often labeled with the term “boost-glide") is
136
launched from the ground by a vertical rocket booster and then glides to its target. Alternatively, the glider and rocket
137
booster could be carried together under a larger, slower aircraft. The rocket booster then brings the glider to a higher
138
altitude and speed from which it can glide to its target.139
Fig. 3 illustrates the historic performance capabilities of all four hypersonic vehicle types in terms of
𝑀𝑚𝑎 𝑥
and
140
𝐴𝑚𝑎 𝑥
. The number of data points for each vehicle type in Fig.
3
are as follows: 8 for Rocket/Air-Launched, 6 for
141
Air-Breather/HTO, 3 for Air-Breather/Air-Launched/Booster, and 3 for Glider/Rocket-Boosted. The encircled areas
142
are not a prediction. Rather, they serve as a visual aid to identify the general areas captured by the data for a given
143
vehicle type. They should be thought of as “vehicle type design spaces.” Since the data is historical, Fig. 3 does not
144
demonstrate what the vehicle types might be capable of but rather what they have been capable of. Still, it provides a
145
gauge for the designer in evaluating the potential capabilities of different vehicle types.146
Payload weight is less helpful in distinguishing vehicle types because most past hypersonic aircraft have been
147
experimental and, as such, have carried little to no payload. Many programs were focused on simply reaching high
148
speeds rather than transporting useful payloads. Certain ranges of
𝑀𝑚𝑎 𝑥
and
𝐴𝑚𝑎 𝑥
in Fig.
3
are not accounted for by
149
historical aircraft data, but future hypersonic vehicles might require performance characteristics that fall within these
150
open spaces. Thus, it is important for the designer to know what vehicle type might be optimal for the full range of
151
performance characteristics in Fig. 3.152
6

0
10
20
30
5 10 15
Maximum Altitude (Amax) [10,000 m]
Maximum Mach Number (Mmax)
Vehicle Type
Air−Breather/Air−Launched/Booster
Air−Breather/HTO
Glider/Rocket−Boosted
Rocket/Air−Launched
Fig. 3 Shaded regions representing the performance capabilities of past hypersonic aircraft distinguished by
vehicle type.
To assess the open space in Fig. 3, a random forest model, implemented with the caret package in R, was used
153
to predict a vehicle type for each combination of
𝑀𝑚𝑎 𝑥
and
𝐴𝑚𝑎 𝑥
. Random forest is an ensemble machine learning
154
algorithm for classification that improves upon a basic decision tree algorithm by combining multiple decision trees
155
[
7
]. The primary limitation in training machine learning models for this research is the lack of training data, as few
156
aircraft have flown at hypersonic speeds. Usually the data are split into two subsets: one for training the model and the
157
other for testing model performance. However, this approach is not realistic for such a small data set. To accommodate,
158
we use a 5-fold cross-validation technique, which is a resampling approach for limited data. Rather than testing the
159
model directly on new, unseen data, which is preferred, we are estimating the model performance. Model accuracy is
160
the percentage of correct predictions averaged across each testing situation. The resulting predictions across the data
161
ranges are displayed in Fig. 4, with a model accuracy of 0.75. The random forest model was used to achieve better
162
performance than a basic decision tree model, which had an accuracy of less than 50%.163
The model predicts the most appropriate vehicle type for a given
𝑀𝑚𝑎 𝑥
and
𝐴𝑚𝑎 𝑥
, but there may be other reasonable
164
vehicle types. It is clear from Fig.
3
that 2 or even 3 vehicle types can cover certain ranges of
𝑀𝑚𝑎 𝑥
and
𝐴𝑚𝑎 𝑥
, but most
165
ranges are dominated by a single vehicle type. Fig.
4
selects a single vehicle type for any combination of
𝑀𝑚𝑎 𝑥
and
166
𝐴𝑚𝑎 𝑥
based on the training data. A few portions of these predictions seem unlikely and probably result from limited
167
training data. The spike up to Mach 25 on the left side of Fig. 4 for Air-Breather/HTO vehicles would be very difficult
168
to achieve, as it requires an air-breather to take off on a runway (horizontal takeoff) and then accelerate to Mach 25
169
without assistance. Additionally, the predictions for Glider/Rocket-Boosted vehicles at lower altitudes and higher Mach
170
numbers seem unlikely as these vehicles reach high altitudes in achieving high Mach numbers. These limitations are not
171
accounted for in the random forest model. As with Fig. 3, Fig. 4 is a gauge for the designer when evaluating types of
172
7

hypersonic aircraft, and the designer will be best served by using both figures together.173
Fig. 4 Random forest model to predict vehicle type based on maximum Mach number and maximum altitude.
B. Initial References Configuration174
1. Regression Models and Uncertainty175
Single variable regression models were used to estimate design parameters with confidence bounds. Specifically,
176
linear and power law models were chosen because these models appear frequently in aircraft design books and reports
177
for similar types of subsonic and supersonic aircraft data [
6
,
8
,
9
] and because they are simple, allowing them to work
178
with relatively few data without overfitting. Uncertainty bounds are captured with the Scheffé band [
10
]. Details are
179
presented for a linear model in [
11
], and the technique has been transformed to capture uncertainty of the power law
180
model. Confidence bands about any regression model are calculated from the Scheffé band weighted for 95% confidence.
181
All confidence bands reported are stated at a 95% confidence level.182
The only uncertainties considered in this analysis are from random errors due to data scatter about the regression
183
model, as well as the propagation of uncertainties due to model random errors within coupled equations and other
184
regression models. This analysis represents a first attempt at quantifying uncertainty for design parameter estimates, so
185
one would expect other errors built into the data used to contribute additional uncertainties. We did not attempt to
186
evaluate or include those.187
Once uncertainty for a parameter has been obtained from a regression model, it can propagate through a defining
188
or physical equation. The technique used here is Taylor series approximations that assign sensitivity weightings to
189
the uncertainty due to each variable, which is described in [
11
]. If
𝑅
has a functional relationship with one or more
190
variables, then191
𝑅=𝑓(𝑥1, 𝑥2, ..., 𝑥𝑛)(1)
8

where
𝑅
is a function of
𝑛
variables [
11
]. The uncertainty in
𝑅
caused by the propagation of uncertainty in the
192
independent variables is given by193
𝑢𝑅=𝜕𝑅
𝜕𝑥1
𝑢𝑥12
+𝜕𝑅
𝜕𝑥2
𝑢𝑥22
+... +𝜕𝑅
𝜕𝑥𝑛
𝑢𝑥𝑛2
(2)
where each partial derivative represents a sensitivity index for that independent variable [
11
]. If a given parameter has
194
both model uncertainty and propagated uncertainty associated with it, these two uncertainties are combined by taking
195
the square root of the sum of the squares.196
2. Weight and Geometry Estimates197
For the purposes of this analysis, gross takeoff weight (
𝐺𝑇 𝑂𝑊
) is divided into three general categories, shown in
198
Eq.
(3)
: empty weight, fuel weight, and payload weight. Each component weight is divided by
𝐺𝑇 𝑂𝑊
to obtain the
199
dimensionless weight fractions in Eq.
(4)
. Dimensionless parameters
𝑓 𝑓
and
𝑠 𝑓
are estimated first before determining
200
values for fuel weight and empty weight.201
𝐺𝑇 𝑂𝑊 =𝑊𝐸+𝑊𝐹+𝑊𝑃(3)
1=𝑠 𝑓 +𝑓 𝑓 +𝑝 𝑓 (4)
Fuel fraction is estimated based on maximum Mach number using Fig. 5. This approach is similar to those in [
8
]
202
and [
9
] for subsonic and supersonic aircraft, where fuel fractions for each phase of the mission profile are estimated.
203
Here the general approach is simplified to just the “climb and acceleration" phase, where fuel fraction can be estimated
204
based on cruise Mach number with a power law relationship [
8
,
9
]. Rather than cruise Mach number, maximum Mach
205
number is used, as it is the more relevant and available parameter for past hypersonic aircraft. The relationship in
206
Fig. 5 makes physical sense: more fuel is required for a given total aircraft weight to achieve the thrust necessary for
207
acceleration to higher Mach numbers. Fuel fraction increases at a decreasing rate because no aircraft can dedicate all
208
the weight to fuel, as captured by the power law model. Uncertainty for
𝑓 𝑓
is estimated from the 95% confidence band
209
in Fig. 5.210
Structural factor is estimated in conjunction with
𝐺𝑇 𝑂𝑊
using two equations. The first describes the physical
211
relationship between
𝐺𝑇 𝑂𝑊
,
𝑊𝑃
,
𝑓 𝑓
, and
𝑠 𝑓
. Solving for
𝑠 𝑓
and substituting in the definition of payload fraction
212
results in Eq. (5).213
𝑠 𝑓 =1−𝑓 𝑓 −𝑊𝑃
𝐺𝑇 𝑂𝑊 (5)
The second equation comes from Fig. 6, which is a regression model for
𝑠 𝑓
as a function of
𝐺𝑇 𝑂𝑊
. Fig. 6
214
9

0246810
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
95% Confidence Band
Fig. 5 Regression model for fuel fraction as a function of maximum Mach number for Rocket/Air-Launched
hypersonic vehicles.
shows that as the total weight of the aircraft increases, the percentage of empty weight decreases at a decreasing
215
rate. Essentially, as
𝐺𝑇 𝑂𝑊
increases, the aircraft increasingly becomes a fuel tank. Eq.
(5)
can be thought of as
216
available structural factor, while Fig. 6 (and the corresponding equation) can be thought of as required structural factor
217
[
8
,
9
]. These two equations are solved simultaneously using an iterative technique to determine estimates for
𝑠 𝑓
and
218
𝐺𝑇 𝑂𝑊
. Uncertainty for
𝑠 𝑓
is estimated from the 95% confidence band in Fig. 6 once estimates for
𝑠 𝑓
and
𝐺𝑇 𝑂𝑊
are
219
established. The estimate for 𝐺𝑇 𝑂𝑊 is assumed to be correct with no uncertainty.220
0.5 1 1.5 2 2.5 3 3.5 4
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
95% Confidence Band
Fig. 6 Regression model for structural factor as a function of gross takeoff weight for Rocket/Air-Launched
hypersonic vehicles.
Empty weight and fuel weight are simply calculated according to Eqs.
(6)
and
(7)
, respectively. The uncertainty in
221
these estimates is determined from the propagation of the uncertainty for 𝑠 𝑓 and 𝑓 𝑓 through the respective equations.222
10

𝑊𝐸=𝑠 𝑓 (𝐺𝑇 𝑂𝑊)(6)
𝑊𝐹=𝑓 𝑓 (𝐺𝑇𝑂𝑊 )(7)
𝑢𝑊𝐸
=𝑢𝑠 𝑓 (𝐺𝑇𝑂𝑊 )(8)
𝑢𝑊𝐹
=𝑢𝑓 𝑓 (𝐺𝑇 𝑂𝑊)(9)
To bridge the gap between weight estimates and geometry, the relationship between
𝑆𝑝𝑙𝑎𝑛
and
𝑂𝑊 𝐸
is utilized,
223
where
𝑂𝑊 𝐸 =𝑊𝐸+𝑊𝑃
. The usefulness of this relationship for hypersonics is discussed in [
3
], and a correlating
224
equation for conceptual air-breathing aircraft is provided in [
12
]. Part of the convergence logic for hypersonic conceptual
225
design discussed in [
3
] includes iterating
𝑆𝑝𝑙𝑎𝑛
to determine
𝑂𝑊 𝐸
and
𝑆𝑝𝑙𝑎𝑛
. A different and much more simple logic
226
is used here for initialization, making use of historical aircraft data and the direct inclusion of payload weight.227
Fig. 7 shows the strong modeling relationship between
𝑆𝑝𝑙𝑎𝑛
and
𝑂𝑊 𝐸
. Lift is linearly proportional to the area of
228
the lifting surface [
13
], which is the planform area. This relationship makes physical sense because when cruising, the
229
downward force of the weight of the aircraft must be counteracted by the upward lift force. Thus, it is not surprising that
230
𝑆𝑝𝑙𝑎𝑛 can be modeled linearly as a function of 𝑂𝑊 𝐸 .231
2 4 6 8 10 12 14
0
20
40
60
80
100
120
95% Confidence Band
Fig. 7 Regression model for planform area as a function of operating weight empty for Rocket/Air-Launched
hypersonic vehicles.
Uncertainty for the
𝑆𝑝𝑙𝑎𝑛
estimate comes from evaluating two error sources: the propagation of uncertainty from
232
𝑂𝑊 𝐸
and the model uncertainty from Fig. 7. First, the uncertainty of
𝑂𝑊 𝐸
is calculated as the propagation of
233
uncertainty from 𝑊𝐸, which is just equivalent to 𝑢𝑊𝐸.234
Second, the uncertainty of
𝑆𝑝𝑙𝑎𝑛
from
𝑢𝑂𝑊 𝐸
propagated through the regression model equation is calculated. This
235
will be (𝑢𝑆𝑝𝑙 𝑎𝑛 )𝑝𝑟 𝑜 𝑝 .236
11

𝑢𝑆𝑝𝑙𝑎 𝑛 𝑝𝑟𝑜 𝑝
=9.04 (𝑢𝑂𝑊 𝐸 )(10)
Third, the uncertainty of
𝑆𝑝𝑙𝑎𝑛
from the model is estimated from the 95% confidence band in Fig. 7. This will be
237
(𝑢𝑆𝑝𝑙𝑎 𝑛 )𝑚𝑜𝑑𝑒𝑙
. Finally, to determine a total uncertainty for
𝑆𝑝𝑙𝑎𝑛
, the two uncertainties are combined as the square root
238
of the sum of the squares.239
𝑢𝑆𝑝𝑙𝑎 𝑛
=𝑢𝑆𝑝𝑙𝑎 𝑛 𝑝𝑟𝑜 𝑝 2
+h𝑢𝑆𝑝𝑙 𝑎𝑛 𝑚𝑜𝑑𝑒 𝑙 i2
(11)
A value for
𝜏
is chosen and can be varied to consider different geometries.
𝜏=0.1
is a reasonable starting point. An
240
important factor is fuel type and its density (and thus volume requirements). For example, an aircraft using hydrogen
241
will probably require a higher value of
𝜏
due to the lower density of hydrogen compared to hydrocarbon fuels. A
242
reasonable value for
𝜏
will also depend on vehicle type. In general, a rocket-powered aircraft will require more volume
243
than an air-breather because it must carry oxidizer.244
Based on the value of 𝜏chosen, total volume is calculated according to Eq. (12) [5].245
𝑉𝑡𝑜𝑡 𝑎𝑙 =𝜏 𝑆1.5
𝑝𝑙𝑎𝑛 (12)
Uncertainty for 𝑉𝑡𝑜𝑡 𝑎𝑙 is calculated as the propagation of uncertainty from 𝑆𝑝𝑙 𝑎𝑛 .246
𝑢𝑉𝑡𝑜𝑡 𝑎𝑙
=(1.5)(𝜏)𝑆𝑝 𝑙𝑎 𝑛 𝑢𝑆𝑝𝑙𝑎 𝑛 (13)
The area ratio
𝐾𝑤
is calculated with Eq.
(14)
, which is from [
3
] and is for a wide range of geometries, vehicle types,
247
and fuel types, including both hydrocarbons and hydrogen. The estimation of
𝐾𝑤
is assumed to be correct with no
248
uncertainty.249
𝐾𝑤
𝜏
=𝑒𝑥 𝑝 [0.081(𝑙 𝑛𝜏)2−0.401(𝑙 𝑛𝜏) + 1.738](14)
The wetted area is calculated with Eq.
(15)
, and uncertainty calculated as the propagation of uncertainty from
𝑆𝑝𝑙𝑎𝑛
.
250
𝑆𝑤𝑒𝑡 =𝐾𝑤𝑆𝑝𝑙𝑎𝑛 (15)
𝑢𝑆𝑤𝑒𝑡
=𝐾𝑤𝑢𝑆𝑝𝑙𝑎 𝑛 (16)
AR is estimated using a regression model with
𝑀𝑚𝑎 𝑥
in Fig. 8. There are several reasons that this trend makes
251
physical sense. Historically, the lifting surfaces (planform areas) of hypersonic aircraft have been increasing, which
252
12

tends to reduce the aspect ratio. This is in part because hypersonic aircraft tend to use the entire bottom surface of
253
the aircraft for lift (from shock wave compression) rather than just two distinct wings. Additionally, achieving high
254
speeds requires reducing drag wherever possible (such as wave drag), likely resulting in tighter aircraft configurations.
255
Uncertainty for 𝐴𝑅 is estimated from the 95% confidence band in Fig. 8.256
0246810
0
2.5
5
7.5
10
12.5
15
95% Confidence Band
Fig. 8 Regression model for aspect ratio as a function of maximum Mach number for Rocket/Air-Launched
hypersonic vehicles.
Wingspan is estimated with Eq.
(17)
, which is the rearranged definition of aspect ratio
[8]
. Uncertainty for
𝑏
is
257
calculated as the propagation of uncertainty from both 𝐴𝑅 and 𝑆𝑝𝑙 𝑎𝑛 .258
𝑏=(𝐴𝑅)𝑆𝑝𝑙𝑎𝑛 (17)
𝑢𝑏=
1
2v
u
u
t(𝑢𝐴𝑅)2𝑆𝑝 𝑙𝑎 𝑛 2+(𝐴𝑅)2𝑢𝑆𝑝𝑙𝑎 𝑛 2
(𝐴𝑅)𝑆𝑝𝑙𝑎𝑛 (18)
Length is estimated based on
𝐺𝑇 𝑂𝑊
using the regression model in Fig. 9. The trend in Fig. 9 makes physical sense
259
as a heavier aircraft would generally be geometrically larger than a lighter aircraft. More specifically, if an increase in
260
𝐺𝑇 𝑂𝑊
corresponds to an increase in
𝑉𝑡𝑜𝑡 𝑎𝑙
, then length might be increased for the aircraft form and the need to reduce
261
drag. Uncertainty for 𝐿is estimated from the 95% confidence band in Fig. 9.262
C. Initial Design Space Exploration263
After obtaining an initial reference configuration, design spaces can be utilized to further assess key aircraft
264
characteristics. It is understood that more than one but not all designs can fulfill a set of mission requirements. The goal,
265
then, is to identify a subset of reasonable solutions, as illustrated in Fig. 10. Eventually, a point design must be decided
266
upon, but for initialization, a collection of reasonable solutions is sought. One method to identify spaces of reasonable
267
design solutions is presented here.268
13

0.5 1 1.5 2 2.5 3 3.5 4
5
10
15
20
25
95% Confidence Band
Fig. 9 Regression model for length as a function of gross takeoff weight for Rocket/Air-Launched hypersonic
vehicles.
Fig. 10 For design initialization, a subset of reasonable design solutions is identified.
Design parameters are treated as interrelated with multivariable regression to consider how changing combinations
269
of those parameters will affect the desired outcomes. With multivariable regression, a performance characteristic is
270
modeled as a function of multiple design parameters (this is the reverse of the previous section, where a design parameter
271
was modeled as a function of a performance characteristic for convenience in calculations).272
In this way, a performance characteristic might be modeled as a function of any number of design parameters. For
273
simplification and visualization, only two design parameters were used to build multivariable regression models for
274
each of the three performance characteristics under consideration:
𝑀𝑚𝑎 𝑥
,
𝑊𝑃
, and
𝐴𝑚𝑎 𝑥
. To maximize the amount of
275
information displayed, each performance characteristic was evaluated for the ranges of design parameter values in the
276
data sets to produce the contour plots in Figs.
14
-
16
. These are inspired by the thermodynamic charts, with lines of
277
constant 𝑀𝑚𝑎𝑥 ,𝑊𝑃, and 𝐴𝑚𝑎 𝑥 as the two design parameters vary.278
The design parameters used in Figs. 14-16 were chosen based on their modeling importance in predicting each
279
performance characteristic. This is captured with a "variable importance" metric in Figs. 11-13, where each relative
280
importance value is scaled to be between 0 and 100. A non-linear, model-independent metric is used to evaluate variable
281
14

importance, where a loess (local weighted regression) smoother is fit and the R
2
statistic is compared to the intercept
282
only null model. Since the variable importance metric is rooted in modeling, assigning appropriate values to important
283
variables does not necessarily cause the desired outcome. Still, the variable importance metric does indicate that there
284
is a strong modeling relationship between the variable and an outcome and that choosing appropriate design parameter
285
values could be important for attaining the desired performance characteristics.286
Fig. 11 Variable importance for design parameters in predicting maximum Mach number for Rocket/Air-
Launched hypersonic vehicles.
Fig. 12 Variable importance for design parameters in predicting payload weight for Rocket/Air-Launched
hypersonic vehicles.
A support vector machine (SVM) algorithm with a polynomial kernel, implemented with the caret package in R, is
287
used for all three models in Figs. 14-16. Missing values in the data were input with bagged trees. To account for the
288
different scales of the data, further preprocessing included centering and scaling the data to improve model performance.
289
15

Fig. 13 Variable importance for design parameters in predicting maximum altitude for Rocket/Air-Launched
hypersonic vehicles.
As with the random forest model for classification, a 5-fold cross-validation technique was used for training and testing
290
each model because of the limited training data. A range of turning parameters were sampled to minimize the RSME
291
(root-mean-square error) for each model. Final R2values for Figs. 14-16 are 0.995, 0.957, and 0.976, respectively.292
The general trend in Fig. 14 is that aircraft flying at higher Mach numbers have higher fuel fractions and lower
293
aspect ratios. This is also indicated by the separate regression models in Figs. 5 and 8. The general trend in Fig. 15
294
is that aircraft carrying heavier payloads tend to have lower aspect ratios and be longer. The greater length is likely
295
because the aircraft would have to be larger in some way to accommodate more weight. Increasing length instead of
296
width could be to reduce form drag by making the aircraft more streamlined. The lower aspect ratio could indicate
297
an increase in planform area to generate more lift for the greater aircraft weight. The general trend in Fig. 16 is that
298
aircraft flying highest are longer with a lower structural factor. The lower structural factor indicates a lighter structure,
299
which results in less extra weight to carry up to altitude. The greater length is likely because the aircraft would have to
300
accommodate more fuel weight to climb up to a higher altitude and thus be larger in some way. As suggested before,
301
increasing length instead of width or height could be to reduce form drag by making the aircraft more streamlined.302
III. Analysis and Results303
An example is provided with procedural steps listed numerically to demonstrate application of the initialization
304
framework. Using a sample design proposal with specific mission requirements, an initial, general estimation of aircraft
305
design and possibilities is obtained. Uncertainty gives bounds outside of which design parameters would be considered
306
unreasonable (or need justification) at this stage.307
16

2
3
4
5
6
7
8
9
1 2 3 4 5 6 7
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Fig. 14 Contour plot with lines of constant maximum Mach number for Rocket/Air-Launched hypersonic
vehicles.
3000
4000
5000
6000
7000
8000
10 12 14 16 18 20 22
1
2
3
4
5
6
7
Fig. 15 Contour plot with lines of constant payload weight [N] for Rocket/Air-Launched hypersonic vehicles.
A. Design Proposal308
The aircraft will be a rocket-powered hypersonic vehicle launched from the air under a larger, slower aircraft (similar
309
to the X-15 [
14
]). After detaching, the aircraft will accelerate to reach a maximum speed of Mach 8 and a maximum
310
altitude of 75,000 m. It will carry a payload of 5,000 N. Overall size restrictions limit the length to 18 m and the width
311
to 12 m.312
B. Application of Framework313
1. Confirm that the proposed vehicle type is a reasonable choice to fulfill mission requirements. The mission
314
requirements for
𝑀𝑚𝑎 𝑥
and
𝐴𝑚𝑎 𝑥
correspond closely to the Rocket/Air-Launched vehicle data in Fig. 3, indicating that
315
it is reasonable for an aircraft of this type to fulfill the requirements. Fig. 4 confirms this conclusion. If that were not
316
17

30000
40000
50000
60000
70000
80000
90000
100000
110000
10 12 14 16 18 20 22
0.3
0.35
0.4
0.45
0.5
0.55
Fig. 16 Contour plot with lines of constant maximum altitude in m for Rocket/Air-Launched hypersonic vehicles.
the case, then this is an opportunity to evaluate whether the proposed vehicle type is reasonable and provide justification
317
for moving forward. Similarly, if the vehicle type was not stated in the design proposal, this step would be to determine
318
which hypersonic vehicle type is best suited to fulfill the mission requirements.319
2. Determine the fuel fraction for a maximum Mach number of 8 from Fig. 5: 𝑓 𝑓 =0.65 ±0.03.320
3. Determine the structural factor and
𝐺𝑇 𝑂𝑊
from Fig. 6 and Eq.
(5)
using a payload weight of 5000 N:
𝐺𝑇 𝑂𝑊 =321
228000 N and 𝑠 𝑓 =0.33 ±0.03.322
4. Determine the empty weight for a 228100 N aircraft from Eq. (6): 𝑊𝐸=75200 ±6840 N.323
5. Determine the fuel weight for a 228100 N aircraft from Eq. (7): 𝑊𝐹=148000 ±6840 N.324
6. Determine the planform area from Fig. 7 by first calculating operating weight empty:
𝑂𝑊 𝐸 =80200 ±6840
N
325
and 𝑆𝑝𝑙𝑎𝑛 =52.0±7.25 m2.326
7. Determine the total volume using Eq. (12) based on an initial value for 𝜏of 0.1: 𝑉𝑡 𝑜𝑡 𝑎𝑙 =37.5±7.85 m3.327
8. Determine the area ratio 𝐾𝑤using Eq. (14) and an initial value for 𝜏of 0.1: 𝐾𝑤=2.2.328
9. Determine the wetted area using Eq. (15): 𝑆𝑤𝑒𝑡 =114 ±16.0m2.329
10. Determine the aspect ratio for a maximum Mach number of 8 from Fig. 8: 𝐴𝑅 =1.3±0.35.330
11. Determine the wingspan using Eq.
(17)
with an aspect ratio of
1.3
and a planform area of 52.0 m
2
:
𝑏=8.22±1.25331
m.332
12. Determine the length from Fig. 9 for a 𝐺𝑇𝑂𝑊 of 228100 N: 𝐿=17.0±1.3m.333
13. Consider how maximum Mach number varies with fuel fraction and aspect ratio according to Fig. 14 along the
334
𝑀𝑚𝑎 𝑥 =8line.335
14. Consider how payload weight varies with aspect ratio and length according to Fig. 15. Note the maximum
336
length restriction of 𝐿=18 m, and based on Fig. 14, 𝐴𝑅 should not be above roughly 3.2.337
18

15. Consider how maximum altitude varies with fuel fraction and aspect ratio according to Fig. 16. As before, there
338
is a maximum length restriction of
𝐿=18
m, and based on Fig. 14, there is a lower limit for
𝑓 𝑓
of roughly 0.62. This
339
requires an upper limit on 𝑠 𝑓 of about 0.38 (this value assumes no payload fraction).340
16. Start making initial sketches with reasonable design parameter values and explore different design possibilities.
341
Steps 13-15 provide guidance on reasonable ranges for certain design parameters. Any sketches must be informed by
342
knowledge of hypersonic aircraft form, which is not explicitly discussed here (see Ref. [
3
]). The initial estimates and
343
design space exploration can then serve as a starting point for the conceptual design process.344
A summary of estimates for design parameters is provided in Table
1
. The mean is the estimated parameter value,
345
while the min and max are the lower and upper ends of the uncertainty range, respectively. The coefficient of variation
346
(COV) is reported as a percentage and is the standard deviation normalized by the mean and multiplied by 100.347
Table 1 Summary of design parameter estimates and ranges.
Parameter Mean Min Max COV Units
𝑓 𝑓 0.65 0.62 0.68 4.6 -
𝐺𝑇 𝑂𝑊 228000 N
𝑠 𝑓 0.33 0.3 0.36 9.1 -
𝑊𝐸75200 68400 82000 9.1 N
𝑊𝐹148000 141000 155000 4.6 N
𝑂𝑊 𝐸 80200 73400 87000 8.5 N
𝑆𝑝𝑙𝑎𝑛 52.0 44.8 59.3 14 m2
𝑉𝑡𝑜𝑡 𝑎𝑙 37.5 26.7 45.4 21 m3
𝑆𝑤𝑒𝑡 114 98.0 130 14 m2
𝐴𝑅 1.3 0.95 1.65 27 -
𝑏8.22 6.97 9.47 15 m
𝐿17.0 15.7 18.3 7.6 m
IV. Discussion348
The example mission requirements fall just outside the range of performance for past Rocket/Air-Launched vehicles
349
in Fig. 3. It is certainly desirable to have more data for referencing and modeling, and this can be added with the
350
development of new aircraft. However, even if the data are sparse, they still provide valuable information about what has
351
already worked for hypersonic aircraft. It also must not be forgotten when assessing vehicle types that, as technology
352
develops, various vehicle types will have expanding capabilities.353
In the example, it was demonstrated that a 228,100 N aircraft with a length of
17.0±1.3
m, a planform area of
354
52.0±7.26
m
2
, and a total volume of
37.5±7.85
m
3
(among other estimates) is a reasonable starting design, with
355
uncertainties providing a confidence level and reasonable range. Uncertainties range from
±4.6
% to
±27
%, which are
356
considered acceptable for this beginning stage of design. While Figs. 14-16 are representative examples, other contour
357
19

plots could be used for other design parameters depending on what the designer is most interested in. More than two
358
design parameters can be modeled together, but this would require a different visualization. All models are only plotted
359
within the available data ranges, which limits the extent of application, and no attempts at extrapolation are made here.
360
While this example and the initialization framework outline are for rocket-powered, air-launched hypersonic vehicles,
361
the framework can theoretically be extended to at least three other types of hypersonic vehicles: Air-Breather/Air-
362
Launched/Booster, Air-Breather/HTO, and Glider/Rocket-Boosted. The approach would be the same but drawing on
363
databases specific to those vehicle types. Part 1 of the initialization framework is not specific to any vehicle type, as it
364
already includes data for four different types of high-speed aircraft. It need not be altered. Parts 2 and 3 would require
365
data and models specific to a given vehicle type. For Part 2, all equations hold true regardless of vehicle type. For Part
366
3, different parameters could be important in design space exploration.367
A concern arises in that the set of mission requirements used for the initialization framework is limited to three at
368
the exclusion of other common performance characteristics. Naturally, the limited requirements translate to limitations
369
in the scope of the design initialization framework. However, this design framework only establishes a low-fidelity
370
starting point, and other requirements can be included as the design progresses, incorporating higher fidelity models.371
Another concern arises regarding the relevance of historical aircraft data when designing new,innovative high-speed
372
aircraft. There are two responses to this. First, this is not a new problem and has long been the case with aircraft design.
373
Aircraft design books, such as those by Nicolai [
9
], Roskam [
15
], and Corke [
8
], make use of historical aircraft data for
374
subsonic and supersonic aircraft. Second, the procedures presented here are for initialization, which is by nature at a
375
low fidelity level. Its goal is to establish a reasonable starting place as a launching pad for the rest of the design process.
376
Starting with what has been known to work, new and innovative designs can be developed.377
V. Conclusion378
To support efforts to develop new hypersonic aircraft, we developed and outlined an initialization framework for the
379
conceptual design of hypersonic vehicles based on historical aircraft data. Hypersonic flight is an important frontier
380
for the future of aircraft design. The framework, data, and models presented in this paper provide unique tools for
381
hypersonic aircraft design not previously presented in the literature. Our new, systematic framework assists the designer
382
in 1) assessing the performance of four types of hypersonic vehicles, 2) establishing basic, initial estimates for weights
383
and geometry, and 3) exploring initial design spaces. It is expressly designed as a first step to be utilized quickly
384
and without any high-fidelity software. While many resources provide historical aircraft data and regression models
385
for subsonic and supersonic aircraft, we provide here data, models, and guidance for the initialization of distinctly
386
hypersonic aircraft designs. Additionally, our framework incorporates machine learning algorithms, explicitly quantifies
387
uncertainty for design parameters, and provide a means for very preliminary design space exploration. Design spaces
388
are generalized as contour plots, applicable to all design scenarios within the data ranges. As new hypersonic designs
389
20

are developed, the models can be informed by greater quantities of training data. The entire framework can conceivably
390
be compiled into a user-friendly computer program if desired. This initialization framework provides rapid, low-fidelity
391
analysis methods that are straightforward for the designer to use in establishing a starting point for the conceptual design
392
of hypersonic aircraft.393
References394
[1] McCullough, D., The Wright Brothers, Simon & Schuster, 2015.395
[2] Anderson, J., Hypersonic and High Temperature Gas Dynamics, American Institute of Aeronautics and Astronautics, Reston,396
VA, 2000.397
[3]
Czysz, P., Bruno, C., and Chudoba, B., Future Spacecraft Propulsion Systems and Integration: Enabling Technologies for
398
Space Exploration, 3rd ed., Springer, 2018.399
[4]
Ingenito, A., Gulli, S., and Bruno, C., “Preliminary Sizing of an Hypersonic Airbreathing Airliner,” Transactions of the Japan
400
Society for Aeronautical and Space Sciences, Aerospace Technology Japan, Vol. 8, No. 27, 2010, pp. 19–28.401
[5] Küchemann, D., The Aerodynamic Design of Aircraft, Pergamon Press, London, 1978.402
[6]
Sieron, T., Fields, D., Baldwin, A., and Adamczak, D., “Procedures and Design Data for the Formulation of Aircraft
403
Configurations,” Tech. rep., Wright Laboratory, August 1993.404
[7]
Shalev-Shwartz, S., and Ben-David, S., Understanding Machine Learning: From Theory to Algorithms, Cambridge University
405
Press, 2014.406
[8] Corke, T., Design of Aircraft, Pearson Education, 2003.407
[9]
Nicolai, L., and Carichner, G., Fundamentals of Aircraft and Airship Design, Volume 1 – Aircraft Design, American Institute of
408
Aeronautics and Astronautics, 2010.409
[10] Scheffé, H., The Analysis of Variance, Wiley Classics Library, Wiley, 1999.410
[11] Figliola, R., and Beasley, D., Theory and Design for Mechanical Measurements, Wiley, 2015.411
[12] Czysz, P., “Hypersonic Convergence, Volume I,” Tech. rep., Air Force Research Laboratory, 2004.412
[13]
Anderson, J. D., Introduction to Flight, 3
rd
ed., McGraw-Hill Series in Aeronautical and Aerospace Engineering, McGraw-Hill,
413
1989.414
[14] Miller, J., The X-Planes: X-1 to X-45, Midland, 2010.415
[15]
Roskam, J., Aircraft Design, Part I: Preliminary Sizing of Airplanes, Design, Analysis and Research Corporation (DARcorpor-
416
tation), 2018.417
21