multidisciplinary design optimization of a two-stage-to

Multidisciplinary Design Optimization of a Two-Stage-to-OrbitReusable Launch Vehicle with

Ethanol-Fueled Rocket-Based Combined Cycle Engines*

Takahiro FUJIKAWA,1)† Takeshi TSUCHIYA,1) and Sadatake TOMIOKA2)

1)Department of Aeronautics and Astronautics, The University of Tokyo, Tokyo 113–8656, Japan2)Kakuda Space Center, Japan Aerospace Exploration Agency, Kakuda, Miyagi 981–1525, Japan

A fully reusable two-stage-to-orbit launch vehicle with ethanol-fueled rocket-based combined cycle (RBCC) engineshas been studied in Japan as a promising option for future space transportation system. In this paper, a conceptual designstudy of such a vehicle is conducted using multidisciplinary design optimization (MDO) techniques in order to clarify atechnology goal for related technology development activities. An MDO framework composed of coupled analysis dis-ciplines (vehicle geometry, mass property, aerodynamics, propulsion, and trajectory) is constructed. In particular, consid-eration is given to the development of a simplified numerical model for evaluating the airframe-propulsion integration thatcan be incorporated into MDO studies, in contrast to costly CFD simulations. Vehicle design and ascent trajectory are thensimultaneously optimized with the aim of minimizing the gross mass of the mated vehicle (booster and orbiter). The grossmass of the obtained optimal design is 581 t for the assumed mission of transporting an 800 kg payload into a low Earthorbit. A detailed inspection of the solution reveals that an external nozzle of the engines enhances not only the propulsionperformance, but also longitudinal static stability of the vehicle during hypersonic flight.

Key Words: Multidisciplinary Design Optimization, Two-Stage-to-Orbit Reusable Launch Vehicle, Rocket-BasedCombined Cycle Engine, Conceptual Design Study, Future Space Transportation System

Nomenclature

¡: angle of attack¤: elevon deflection (positive downward)z: design variables for booster airframe

CL: lift coefficientCD: drag coefficient

CM;ref : pitching moment coefficient about reference pointa: polynomial coefficient in the approximation of aero-

dynamic coefficientM: pitching moment about center of mass

Scapt: inlet flow capture area¸: throttling parameterp: static pressureq: dynamic pressure

Ma: Mach number£: heat capacity ratio

CT : ram thrust coefficientIsp: specific impulse

O=F : oxidizer-to-fuel mass mixture ratioT: thrust

SubscriptsR: embedded rocket in RBCC engine

max: maximum

sur: surrogate model1: free-stream conditionin: inlet inflow conditione: exit condition of RBCC combustorA: airframe

eng: RBCC engineext: external nozzletot: total

1. Introduction

Quite high levels of cost efficiency, operability, and reli-ability are indispensable for future space transportation sys-tem in order to attract potential users of outer space and topromote the growth of the space transportation market. Sinceit is dubious that these requirements can be satisfied by im-proving expandable launch vehicles, reusable launch ve-hicles (RLVs) or space planes have been studied for years.An essential effort for realizing RLVs is to perform their con-ceptual design studies early, and thereby to clarify systemgoals and associated key technologies. At the Japan Aero-space Exploration Agency (JAXA), the feasibility of fully re-usable two-stage-to-orbit (TSTO) systems is currently beinginvestigated as a long-term goal (2030s or later).1) Under atentative mission plan of transporting eight persons or an800 kg payload into a low Earth orbit, design studies oftwo types of booster-stage vehicles equipped with differentpropulsion systems (rocket engines or airbreathing engines)are underway.

This paper focuses on a horizontal takeoff and landing

© 2017 The Japan Society for Aeronautical and Space Sciences+Received 18 September 2015; final revision received 11 August 2016;accepted for publication 3 October 2016.†Corresponding author, [email protected]; Presently,Department of Mechanical and Control Engineering, Kyushu Institute ofTechnology, Fukuoka 804–8550, Japan

Trans. Japan Soc. Aero. Space Sci.Vol. 60, No. 5, pp. 265–275, 2017

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RLV with airbreathing engines, which is one of the idealtypes of space transportation system that would achieve air-craft-like reliability and operability. Among some airbreath-ing propulsion options, a rocket-based combined cycle(RBCC) engine2) is selected here. The RBCC engine is thecombination of a dual-mode ramjet/scramjet engine androcket engines that are embedded in it. A characteristic ofthe RBCC engine is that it can be operated from sea levelto outer space with superior performance to conventionalrockets. In this paper, ethanol is adopted as the fuel of theRBCC engine instead of cryogenic liquid hydrogen becauseit can lead to a compact and easily handled vehicle. In addi-tion, it is assumed that the engine is operated with relativelylow combustion pressure aiming to attain high reliability andto prolong operational life.

Conceptual design studies of a TSTO RLV with ethanol-fueled RBCC engines are challenging tasks, and ongoingstudies are still in the early phases.3–6) Design problems ofsuch airbreathing-powered RLVs contain complex interac-tions among design disciplines, which makes them difficultto be handled from the viewpoints of solution approachesand computational cost. For example, the substantial interde-pendence of vehicle design and flight trajectory design orig-inates from the fact that launch vehicles do not have a cruis-ing state unlike aircraft. This means that a steady-state designpoint for aerodynamic shape and propulsion system cannotbe defined. Besides, forebody pre-compression and after-body exhaust expansion of the propulsion system are relatedto free-stream conditions and airframe geometry in highlynonlinear manners. Because of such dependence of the per-formance of airbreathing engines on flight conditions and air-frame design, vehicle sizing based simply on velocity incre-ments is unsuitable in contrast to rocket vehicles.

In this paper, a multidisciplinary design optimization(MDO) approach is employed so as to overcome the fore-going difficulties. MDO is a design optimization frameworkwhere numerical optimization methods are applied to the de-sign problem of a system composed of multiple interactingdisciplines. MDO techniques have had significant successesin conceptual design studies of RLVs.7–10) Coupled analysisdisciplines considered in this paper are vehicle geometry, ve-hicle mass property, propulsion system, aerodynamics, andflight trajectory. With the MDO architecture constructed, op-timization is executed to find the vehicle design with the min-imum gross mass. Major modifications from a previous re-search by the authors6) include the implementation of anairframe-propulsion integrated analysis. This enables moreaccurate performance evaluation of the RBCC engine con-sidering its interaction with the airframe. Additionally, inthe flight trajectory analysis based on point-mass dynamics,some rigid body characteristics of the vehicle (i.e., pitchingtrim and longitudinal static stability) are also evaluated whiletaking the influence of RBCC engines into account.

The remainder of this paper is organized as follows:Section 2 describes an MDO methodology for the TSTORLV with RBCC engines. Optimization procedures and nu-merical models are explained. In Section 3, the optimal solu-

tion obtained is presented, and detailed inspections are dis-cussed. Finally, Section 4 summarizes this paper andsuggests some future work.

2. A Multidisciplinary Design Optimization Methodol-ogy for a TSTO RLV with RBCC Engines

The design objective is the minimization of the gross massof a mated vehicle (booster and orbiter) that transports an 800kg payload into orbit. While the primary scope of this paperis the conceptual design of the booster vehicle, the perfor-mance of the overall TSTO system cannot be evaluated withthe booster stage alone. Therefore, the scaling of the orbitervehicle and the trajectory design of the orbiter are conductedas well.2.1. Optimization procedures

An MDO framework for an RBCC-powered TSTO RLVis constructed as outlined in Fig. 1. Numerical analysis dis-ciplines in this framework are broken up into vehicle geom-etry and mass property, aerodynamics, propulsion, and flighttrajectory. Design variables are composed of sets of variablesthat specify vehicle design, vehicle performance, and flighttrajectory, respectively, as gathered in Table 1. It is notedthat an optimization problem arising from this MDO frame-work is formulated as an augmented optimal control prob-lem. By appending static variables and static constraints re-garding vehicle design and vehicle performance to acontinuous-time optimal control problem of the vehicle, con-current design optimization of vehicle and flight trajectory isachieved. Using this formulation, there is no nested iterationloop in the solution procedures, and all the objective and con-straints are managed by a single system-level optimizer.Such an MDO architecture is categorized into an all-at-onceapproach, which is known to have stable and rapid conver-gence, and to be free from convergence errors between cou-pling variables in different disciplines.7) For the optimizer, anoff-the-shelf solver named SNOPT,11) which stands forsparse nonlinear optimizer, is employed. SNOPT is an imple-mentation of a sequential quadratic programming (SQP)algorithm that exploits the sparsity in the constraint

Thrust, Isp, O/F,Ext. nozzle performance

Aerodynamics

Trajectory

Propulsion

Dry mass, Dry CM position,Tank vol., Prop. CM position

Scapt, Ext. nozzle geometry

Lift, Drag, Pitching moment

Optimizer(SQP method)

Objective value (vehicle gross mass),Constraint values

Design variables

Reference area,length, and point

Vehicle geometry& Mass property

Fig. 1. Overview of an MDO framework.

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Jacobian,11) and it is suited to the present MDO formulationwhose constraint Jacobian is highly sparse.2.2. Vehicle geometry and mass property analysis

In this analysis discipline, the geometric design of thebooster and that of the orbiter are defined by the design var-iables. Then, the dry mass of the vehicles, volume of thetanks, and center of mass of the vehicles and propellantsare calculated.

The basic configuration of the booster is depicted inFig. 2. In addition to the constraints enumerated in Table 1,some inequality constraints are imposed so that external geo-metries of the airframe, tanks, and engines are successfullydetermined by the design variables. Five RBCC enginesare installed on the undersurface of the fuselage. The flatundersurface of the forebody is employed to supply uni-formly pre-compressed airflow into the engines, and that ofthe afterbody acts as an external nozzle. The airfoils of themain wing and the vertical tail wing are NACA0005, andthe taper ratio of the main wing is fixed at 0.15. Ninety per-cent of the exposed span from the wing tips and 30% chordfrom the trailing edge of the main wing are used as elevons.Right and left elevons are deflected in the same direction and

with identical angles (positive downward) to serve as eleva-tors. The tail wings are scaled so that their total area equals10% of the main wing area. Two integral tanks of ethanolfuel (fore and aft) and cylindrical tanks of liquid oxygen(LOX) are located inside the fuselage. Some control overthe position of the center of mass is possible during the flight

Table 1. Design variables.

Parameter Unit Associated constraints

Vehicle designBooster Length of the fuselage, bl m 30 � bl � 60

Height of the upper fuselage, bh m 0:01 � bh=bl � 0:05

Inclination of the forebody undersurface, bwed deg 3:0 � bwed � 8:0

Angle of the external nozzle, bext deg 3:0 � bext � 20:0

Length of the forebody, blf m 0:3 � blf=bl � 0:6

Width of the forebody tip, bwf m 0:5 � bwf=½ð5=1:81Þeh� � 0:8

Leading-edge position of the exposed wing, w0 m 0:25 � w0=bl � 0:75

Root chord length of the exposed wing, wchrd m 0:25 � wchrd=bl � 0:75

Sweepback of the wing leading-edge, w� deg 45 � w� � 70

(Forward, Backward) end of the fore ethanol tank, (teaf0; teaff ) m 5:0 � teaf0 � teaff(Forward, Backward) end of the LOX tank, (tlo0; tlof ) m teaff þ 0:5 � tlo0 � tlof(Forward, Backward) end of the aft ethanol tank, (teaa0; teaaf ) m tlof þ 0:5 � teaa0 � teaaf � bl � 0:5

Height of RBCC engines, eh m 0:2 � 5:96eh=bl � 0:5

Maximum thrust of RBCC embedded rocket engines, TR,max kN –

Orbiter Length of the fuselage, ol m 10 � ol � 30

Backward end of the cabin, cbf m 0:25 � cbf=ol � 0:9

Rocket engine thrust kN –

Vehicle performanceBooster Maximum axial acceleration G –

Maximum normal load factor G –

Maximum dynamic pressure, qðbÞmax kPa qðbÞmax � 50

Maximum exerted thrust kN –

Gross mass t –

Orbiter Maximum axial acceleration G –

Maximum normal load factor, lfðoÞmax G 2:5 � lfðoÞmax

Maximum dynamic pressure, qðoÞmax kPa qðoÞmax � 50

Maximum exerted thrust kN –

Gross mass t –

Flight trajectoryBooster Parameterized state variables see Section 2.5.

Parameterized control variables see Section 2.5.Switchover times between RBCC engine modes s –

Staging time s –

Orbiter Parameterized state variables see Section 2.5.Parameterized control variables see Section 2.5.Terminal time s –

LOX tanks Ethanol tankEthanol tank

Fig. 2. Basic configuration and design parameters of the booster.


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by adjusting the ratio of consumption of ethanol fuel betweenthe fore and aft tanks. The volume of the integral tanks is cal-culated by integrating the fuselage cross section between theforward end and the rear end, times a volume efficiency con-stant of 80%. It is assumed that the center of mass of the pro-pellant is identical to that of the corresponding tank.

The lifting body configuration shown in Fig. 3 is adoptedto the orbiter based on Fujii et al.1) Airframe geometry opti-mization is not conducted, and rather, only scaling is per-formed in this paper. It is assumed that every 4.3m3 of cabinvolume can accommodate a payload that weighs 100 kg. Theorbiter is loaded onto the upper surface of the booster at sucha position that the center of mass does not move in theaxial direction just before and just after the separation ofthe orbiter. A constraint on the width of the booster fuselageand that of the orbiter is enforced to guarantee the mountabil-ity of the orbiter.

The dry mass and the center of mass of these vehicles arecalculated using HASA,12) a statistical relationship for esti-mating the mass of hypersonic vehicles. Cold structures withaluminum alloy are assumed. An input to HASA consists ofvehicle geometric parameters such as wetted area, maximumloads such as maximum dynamic pressure, and gross mass.These input values are present in or calculated from the de-sign variables. The mass of the RBCC engines is calculatedby adding that of embedded rockets (thrust-to-mass ratio ofthe rocket engine is assumed to be 50) and that of the du-al-mode flow-pass. The structure mass of the flow-pass is es-timated with the statistical relation obtained from FEM-based structural analyses.3) The mass of the thermal protec-tion system is computed assuming a constant areal densitybased on Tsuchiya and Mori.8) The resultant gross mass(i.e., the summation of the calculated dry mass and propellantmass before takeoff) must be consistent with the gross massvalue specified in the design variables. This consistency con-dition is enforced as a constraint in the MDO problem.2.3. Aerodynamic analysis

In the aerodynamic analysis, aerodynamic characteristicsof the booster, the orbiter, and their mated configurationare calculated when the vehicle shape and flight conditionsare specified.

In order to calculate the trajectory of the launch vehicle,aerodynamic properties for a wide range of flight conditionsare needed. Considering this requirement along with the re-quired fidelity level in the conceptual design study, the fol-lowing two types of engineering-level CFD methods are uti-

lized depending on the flow speed. In subsonic or supersonicconditions (Ma1 < 2:0), the A502 PAN AIR panel code,13)

whose governing equation is a linearized potential flow equa-tion with compressibility correction, is employed. In hyper-sonic conditions (Ma1 � 2:0), the tangent cone method14)

and Prandtl-Meyer expansion flow theory are applied tothe windward and leeward regions of the vehicle surface, re-spectively. Although this hypersonic aerodynamic character-ization method is not based on any governing equations, itsaccuracy is adequate for conceptual design purposes. In boththe flow-speed cases, the base pressure of the fuselage isestimated separately with an empirical method,15) and theskin friction coefficient is calculated using van Driest’sformula.16) The external nozzle ramp is excluded when com-puting the aerodynamic coefficients of the booster becausethe forces acting on the ramp are considered in the air-frame-propulsion integrated analysis in Section 2.4. In themated vehicle configuration, it is assumed that the airflowhits the orbiter along its body axis after the free-stream flowpasses by the booster upper surface.

For the optimization computation, the surrogate model ofthe above CFD analysis is built based on Yokoyama et al.9)

in order to mitigate the computational burden and to enhancenumerical stability. The surrogate aerodynamic coefficientsof the booster are the functions of ¡, ¤, Ma1, and z; that is,

Cf�g � Cf�g; surð�; �;Ma1; zÞ; ð1Þwhere f�g represents L, D, orM; ref. In order to train the sur-rogate models, 200 sample points for z are made in advancevia a design-of-experiments algorithm.9) This is a kind ofspace-filling design method, and it generates sample pointsthat uniformly fill the constrained design space. In additionto the sample points, 200 test points for cross-validationare prepared using random sampling. Then, the aerodynamicanalysis is conducted on these sample and test points underthe conditions of ¡, ¤, and Ma1 shown in Table 2 (45 casesfor each z).

When z is provided during optimization, the aerodynamiccoefficients under the sampled conditions of ¡, ¤, and Ma1are calculated using radial basic function (RBF) networks17)

with the Gaussian basis function in the following manner:

Cf�g; sur½ð�; �Þi;Ma1j; z� ¼X200k¼1

�i;j; k exp � jjz� zkjj2� �2

�i;j

� �2" #

;

i ¼ 1; . . . ; 5; j ¼ 1; . . . ; 9: ð2ÞHere, �i;j; k and �i;j are the fitting parameters in the RBF net-works that are optimized beforehand based on the input/re-sponse of aerodynamic computations at sample/test pointsby means of simulated annealing. Subsequently, the aerody-

Table 2. Conditions in aerodynamic analysis.

Parameter(s) Sample values Unit

Ma1 0:2; 0:6; 0:9; 1:2; 3:0; 5:0; 7:0; 10:0; 15:0 –

ð�; �Þ (0, 0), (5, 0), (10, 0), (5, ¹10), (5, 10) (deg, deg)

LOXtank

EthanoltankCabin

Fig. 3. Basic configuration and design parameters of the orbiter.


268©2017 JSASS

namic coefficients are interpolated with respect to Ma1 us-ing RBF networks with the multiquadric basis function asfollows:

Cf�g; sur½ð�; �Þi;Ma1; z�

¼X9j¼1

� 0i;j Ma1 �Ma1j

� �2þ �0i

� �2h i1=2; i ¼ 1; . . . ; 5;

ð3Þ

where � 0i;j are other fitting parameters that are calculated

from Cf�g; sur½ð�; �Þi;Ma1j; z�; i ¼ 1; . . . ; 5; j ¼ 1; . . . ; 9

with slight computation cost, and �0i ¼ 0:01. The aerody-

namic coefficients at the combinations of ¡ and ¤ shown inTable 2 are obtained. Finally, the surrogate aerodynamic co-efficients are constructed as second-order polynomials interms of ¡ and ¤ as follows:

Cf�g; surð�; �;Ma1; zÞ¼ að0;0Þf�g þ að1;0Þf�g �þ að2;0Þf�g �2 þ að0;1Þf�g �þ að0;2Þf�g �2; ð4Þ

where the polynomial coefficients, að0;0Þf�g ; . . . ; að0;2Þf�g , are com-puted via the least squares fit of Cf�g; sur½ð�; �Þi;Ma1; z�;i ¼ 1; . . . ; 5.

The above procedures yield surrogate models whose coef-ficients of determination are 0.988 on average and 0.968 inthe worst case, which is an acceptable prediction error.2.4. Airframe-propulsion integrated analysis

In the propulsion analysis, thrust, Isp, O=F , and externalnozzle performance are calculated when the engine design,the booster airframe, and flight conditions are provided.

Ethanol-fueled RBCC engines, whose conceptual image isillustrated in Fig. 4, are installed on the booster. RBCCengines are operated in ejector-jet (i.e., ducted rocket), ram-jet, and scramjet modes, successively in that order. Ejector-jet and scramjet modes are operated by keeping the outputof embedded rockets high (rocket chamber pressure is6MPa). In ramjet mode, on the other hand, the rocket cham-ber pressure is reduced to 0.6MPa in order to attain better Ispowing to airbreathing effects. The best acceleration perfor-mance is achieved by properly switching these operatingmodes depending on flight conditions.

The detailed geometry of the engine flow-pass is fixed: theoverall length-to-height and height-to-width ratios of an en-gine unit are 5.96 and 1.81, respectively, and the throat-to-in-let area ratio is 0.2.3) The flow-pass is scaled based on the en-gine height included in the design variables, and themaximum thrust of the embedded rocket, TR,max, is also var-iable. An airframe-propulsion integrated analysis, whoseoverview is shown in Fig. 5, is conducted. The analysis iscomposed of three calculation steps: pre-compression,RBCC engine, and external nozzle. For simplicity, influenceof engine operation on the flow field around the airframe dur-ing the subsonic ejector-jet mode is neglected.

(1) Pre-compression analysisA shock wave generated from the booster nose before the

undersurface is equivalent to an oblique shock wave past atwo-dimensional wedge whose angle is ð�þ bwedÞ. Equa-

tions describing oblique shocks18) are employed, and inlet in-flow conditions to RBCC engines are obtained.

(2) RBCC engine analysisThe performance dataset of the RBCC engine at a set of

representative inflow conditions (Main and qin) and TR=Scaptis computed in advance with a quasi-one-dimensional analyt-ical method.3) It is assumed that qin is bounded to [10,100] kPa. When TR=Scapt is varied, the engine thrust-to-massratio in a static condition and Isp in some operating condi-tions change as shown in Fig. 6. A larger embedded rocketrelative to the flow-pass gives a higher thrust-to-mass ratio,while it decreases Isp except in scramjet mode. Then, surro-gate models of responses (i.e., CT , Isp, O=F , Mae, pe, and�e) for each engine operating mode are constructed usingan RBF network with the multiquadric basis function. Inejector-jet and scramjet mode, the thrust of the RBCC enginealone, T, is calculated in the following manner:

T ¼ CT Scapt þ TR; ð5Þwhere

TR ¼ TR,max �: ð6ÞIn ramjet mode, on the other hand,

Fig. 4. A conceptual image of an RBCC engine.4)

Geometry ofext. nozzle

Forces acting on the nozzle

Oblique shock equation

RBCC engine model

Q1D isentropic flow equation

Fig. 5. Overview of airframe-propulsion integrated analysis.

10

15

20

25

30

Thr

ust−

to−

mas

s ra

tio

Engine thrust−to−mass ratio in a static condition (ejector−jet mode, = 2, = 50 kPa)

(ramjet mode, = 4, = 50 kPa)

(scramjet mode, = 7, = 50 kPa)

50 100 150 200 250 300 350 400 450200

300

400

500

600

[s]

[kN/m2]

Isp Main qin

Isp Main qin

Isp Main qin

TR/ Scapt

Isp

Fig. 6. Effects of TR=Scapt on RBCC engine performance.


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T ¼ CT Scapt: ð7ÞThrottling parameter, ¸, is allowed to vary within a range[0.5, 1] only in scramjet mode.

(3) External nozzle analysisThe pressure on the nozzle ramp surface is estimated based

on the following numerical model3):� The exhaust gas of the engine is expanded quasi-one-dimensionally along a flow-pass surrounded by the nozzleramp and an extended line from the engine cowl (thedashed line shown in Fig. 5) in an isentropic manner.

� Under an over-expansion condition, the exhaust flow goesalong the nozzle ramp with static pressure identical to theambient pressure.The force and pitching moment acting on the external noz-

zle are obtained by integrating the calculated flow pressureover the entire ramp surface after the free-stream pressureis subtracted. In order to avoid iterative solutions of nonlin-ear isentropic flow equations during optimization computa-tion, surrogate models of the nozzle performance are con-structed beforehand using an RBF network with themultiquadric basis function. Sample points for training theRBF network surrogate models are prepared via an opti-mized Latin hypercube sampling algorithm.19)

The orbiter is propelled by a rocket engine mounted on thebase of the fuselage. Ethanol fuel and LOX are employed,and vacuum Isp, engine thrust-to-mass ratio, and O=F are320 s, 75, and 1.6, respectively.1) Effective thrust and Isp

are computed considering the effect of atmospheric pressure.It is assumed that the thrust of the rocket engine can be throt-tled without affecting Isp.2.5. Flight trajectory analysis

In the flight trajectory analysis, it is assumed that thetrajectory is restricted to a vertical plane, and two-degree-of-freedom dynamics is employed. A static atmospheremodel is built based on U.S. standard atmosphere 1976.20)

In the context of optimal control, state variables are com-posed of altitude, velocity, flight path angle, remainingethanol fuel mass in the fore tank, that in the aft tank, andremaining LOX mass. Control variables consist of angle ofattack, elevon deflection angle, throttling parameter, andthe consumption ratio of ethanol fuel between the fore andaft tanks. A flight plan is described by the following threephases:

(1) Takeoff phaseThe mated vehicle takes off horizontally from a runway at

sea level. No particular launch site is assumed in this paper.The RBCC engines are operated with the maximum thrust,and angle of attack for liftoff is 15 deg. A takeoff analysis21)

is conducted, and the following quantities are computed:takeoff velocity, takeoff distance (ground roll), and the pro-pellant mass consumed during the takeoff phase. The calcu-lated takeoff velocity and distance must be no more than170m/s and 4,000m, respectively.

(2) Mated vehicle ascent phaseThe mated vehicle is accelerated by the RBCC engines.

The rocket engine mounted on the orbiter is not used. Angle

of attack and elevon deflection are bounded to [¹5, 15] degand [¹10, 10] deg, respectively. Elevon deflection angle islimited, because a large deflection can lead to flow separationand re-attachment, causing local high heat flux and elevon ef-fectiveness loss. Axial acceleration, normal load factor, dy-namic pressure, and exerted thrust must not exceed the cor-responding design limits (see Table 1). Since the TSTORLV investigated is a manned transportation system, totalacceleration (the composition of axial acceleration andnormal load factor) is restricted to 4.0G maximum. At theterminal time of this phase, the orbiter is separated fromthe booster.

This flight phase is further divided into three sub-phaseswhere the RBCC engines are operated in different modes(ejector-jet, ramjet, and scramjet modes in this order). Ramjetmode is available when the engine inflow Mach number,Main, is 3.0 (the designed start Mach number of the inlet)or higher. Switchover times between engine operating modesare included in the design variables, and they are optimizedto achieve the best acceleration performance.

(3) Orbiter ascent phaseThe rocket engine on the orbiter is ignited immediately

after the staging. Angle of attack is bounded to [0, 30] deg,and the throttling parameter must be within a range[0.5, 1]. Axial acceleration, normal load factor, dynamicpressure, and thrust must not exceed their corresponding de-sign limits for the orbiter. The total acceleration is limited tono more than 4.0G.

After the orbiter reaches an altitude of 100 km or higherand the rocket engine is cut off, the trajectory computationis performed based on elliptic orbit equations instead ofequations of motion. At the apogee of this coasting trajec-tory, the orbiter is accelerated again, and it is injected intoa circular orbit at 350 km above sea level. The propellantmass needed for this apogee acceleration is calculated by ap-proximating it as an impulsive acceleration whose Isp is320 s. In addition to the propellant consumed before the orbitinsertion, some propellant is reserved for on-orbit and de-or-bit operations whose velocity-change requirement is as-sumed to be 250m/s in total.

The return trajectory of the booster after the staging andthat of the orbiter after de-orbit are not considered in this pa-per.

During the mated vehicle ascent phase, a pitching trimcondition and a longitudinal static stability condition are im-posed in order to ensure the flyability of the vehicle. The trimcondition is expressed as follows:

Mtot :¼ MA þMeng þMext ¼ 0: ð8ÞAdditionally, the following equation describes the non-neg-ative static stability:

@Mtot

@�¼ @MA

@�þ @Meng

@�þ @Mext

@�� 0: ð9Þ

Here, pitching moment in the above equations is calculatedabout the center of mass, which moves during the flight. Itis noted that conventional stability analysis based on the po-


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sitions of stability neutral point and center of mass is invalid,because the magnitude of moment produced by RBCC en-gines changes depending on aerodynamic attitude. In orderto calculate the value of @Mtot=@� accurately, analytical dif-ferentiation of RBF surrogate models and automatic differen-tiation22) of the nose shock wave relations are implementedin the MDO framework.

As explained in Section 2.1, the MDO problem for theTSTO RLV with RBCC engines in this paper is formulatedas an augmented optimal control problem. In order to discre-tize state variables and control variables, the Legendre-Gausspseudospectral method23) with adaptive mesh refinement24)

is employed. Consequently, these time-dependent variablesare parameterized by the values at a finite number of tempo-ral collocation points, and continuous-time equations of mo-tion and path constraints are transcribed into static con-straints. It should be mentioned that the MDO problemconsists of large numbers of design variables and constraintswhen it becomes a good approximation to the original con-tinuous-time problem. The number of collocation pointsand their distribution are automatically and adaptively deter-mined using the mesh refinement algorithm so that a user-de-fined discretization error tolerance is satisfied.

3. Results and Discussion

3.1. Optimization processThe termination criterion of the numerical optimization

using SNOPT is that the maximum residual of the Karush-Kuhn-Tucker optimality conditions and the largest constraintviolation are both no more than 1:0 10�8, which is a more-than-adequate level for engineering purposes The basic con-figuration of the booster shown in Fig. 2 (fuselage length is35.0m) is used as the initial point for starting optimization.The optimization computation takes about 15 minutes usinga Windows 7 machine with an Intel Core i7-4930K CPU and32GB RAM.3.2. Optimal vehicle design and flight trajectory

The specifications of optimal design obtained are shown inTable 3. The gross mass of the mated vehicle, the design ob-jective to be minimized, is 581 t. This is smaller than themaximum takeoff mass of an An-225 aircraft (600 t25)),which may indicate that there is a realistic possibility inthe horizontal takeoff of RLVs of this scale. Among thebooster-vehicle components, the mass of the landing gear oc-cupies the largest portion, followed by that of the RBCC en-gines.

At JAXA, a fully reusable TSTO launch vehicle with eth-anol-fueled rocket engines is under a conceptual designstudy. The vertical lift-off mass of the rocket-powered sys-tem is 788 t,1) so the RBCC-powered RLV designed in thispaper has smaller gross mass. Of course, a simple compari-son is rather meaningless since the underlying design as-sumptions and numerical models differ substantially.

Details of the optimal vehicle design are shown in Table 4and Fig. 7. Since the orbiter is not much smaller than thebooster, aerodynamic interference between the vehicles must

be considered in more detail in future work. The optimal valueof TR=Scapt in the RBCC engine design is 269 kN/m2. Theoptimal booster shape has the following notable characteris-tics in comparison to the basic configuration shown in Fig. 2.The booster fuselage has the minimum width required to loadthe orbiter on its upper surface. Since the numerical model ofthe external nozzle in Section 2.4 always produces non-neg-ative thrust, the fuselage base of the booster is fully used asthe external nozzle. The inclination of the forebody undersur-face is at the upper bound of the design variable. This is be-cause the improvement of the propulsion efficiency due tostronger pre-compression outweighs the increase in the dragof the airframe. The main wing is located at the rear end of

Table 3. Specifications of the optimal solution.

Parameter Unit Booster Orbiter

Payload mass kg 800.0Mated vehicle gross mass t 580.9Takeoff velocity m/s 170.0Takeoff distance m 3044.4

Maximum axial acceleration G 2.4 2.4Maximum normal load factor G 1.4 2.5Maximum dynamic pressure kPa 49.7 49.7Maximum exerted thrust kN 4420.0 353.6

Mass Fuselage t 9.3 2.9Main wing t 7.9 0.0Tail wing t 1.7 0.6Tanks t 2.3 0.6Thermal protection system t 4.9 2.3Landing gear t 30.4 0.4RBCC engines t 14.7 0.0Rocket engine t 0.0 0.5Thrust structure t 2.8 0.9The others t 11.5 2.5

Dry gross t 85.5 11.4

Ethanol t 192.2 13.7LOX t 256.1 21.9

Gross t 533.8 47.1

Table 4. Details of vehicle design in the optimal solution.

Parameter Unit Value

Booster Fuselage length, bl m 38.4Fuselage upper height, bh m 1.0Forebody undersurface inclination, bwed deg 8.0External nozzle angle, bext deg 14.6Forebody length, blf m 13.9Forebody tip width, bwf m 4.2Exposed wing leading-edge position, w0 m 20.6Exposed wing root chord length, wchrd m 17.8Wing leading-edge sweepback, w� deg 53.4Fore ethanol tank end (front), teaf0 m 5.0

(back), teaff m 12.9LOX tank end (front), tlo0 m 13.7

(back), tlof m 27.8Aft ethanol tank end (front), teaa0 m 28.3

(back), teaaf m 37.9Height of the RBCC engines, eh m 2.2RBCC embedded rocket thrust, TR,max kN 3652.1

Orbiter Fuselage length, ol m 18.8Back end of the cabin, cbf m 11.2Rocket thrust kN 353.6


271©2017 JSASS

the fuselage. This moves the neutral point for longitudinalstatic stability of the airframe rearward and ensures the stabil-ity up to a higher Mach number.

The flight trajectory of the optimal solution is shown inFig. 8. After horizontal takeoff, the mated vehicle acceleratesand climbs with the RBCC engines at full throttle and with alarge angle of attack. Since the vehicle thrust-to-weight ratioat takeoff is 0.63, which is less than 1, the lift force must playa major role in the takeoff. During ejector-jet mode opera-tion, flight dynamic pressure increases, decreases, and in-creases again. In contrast to a TSTO RLV with hypersonic

turbojet engines,8) the optimal flight trajectory does not havea dive phase in the transonic region. This dissimilarity can beattributed to the fact that the thrust-to-mass ratio of theRBCC engine is higher than that of the hypersonic turbojetengine. Therefore, the optimal RBCC-powered vehicle hasenough thrust performance to pass the transonic drag risewithout a dive maneuver. The engine operating mode isswitched over to ramjet mode at a flight Mach number of3.6. This corresponds to the condition where the engine in-flow Mach number, Main, reaches 3.0, and ramjet mode be-comes available. Ramjet mode is operated at the maximumdynamic pressure prior to 306 s, at which point the dynamicpressure begins decreasing. Subsequently, the RBCC en-gines are switched to scramjet mode when the vehiclereaches Mach 5.8. During acceleration in scramjet mode,the engines are gradually throttled down. Finally, the vehiclebegins a steep climb by raising its angle of attack.

In a previous conceptual design study of the TSTO RLVwith RBCC engines,6) in which the airframe-propulsion in-teraction is neglected, most of the optimal flight trajectoryof the mated vehicle lies on the upper dynamic pressure limit.On the other hand, a quasi-one-dimensional model for the ex-ternal nozzle flow is incorporated in the present study, andthe resulting optimal trajectory takes on a more intricate be-havior. Since thrust augmentation by the external nozzle in-

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Fig. 8. Optimal flight trajectory.

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Fig. 7. Optimal vehicle shape.


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creases as the ambient pressure drops, the mated vehicle in-creases altitude without flying along the maximum dynamicpressure around 100 s in order to enhance the overall per-formance of the RBCC engines.

When the propellant in the booster is used up, the orbiter isseparated from the booster. At that moment, the Mach num-ber and dynamic pressure are 12.2 and 6.3 kPa, respectively.The orbiter ascends quickly to an altitude of 100 km withmaximum rocket thrust, and then it continues to acceleratewhile descending slightly. At the terminal time, the orbiteris on an elliptic orbit with an apogee of 350 km above sealevel.

Until 135 s after takeoff, ethanol fuel is consumed onlyfrom the aft tank, followed by the consumption almost onlyfrom the fore tank until 470 s. After that, the fuel is expendedonly from the aft tank again. This switching strategy in thefuel consumption maximizes the effectiveness of the elevonswhile attaining non-negative static stability.3.3. Effect of external nozzle

Time histories of the performance of the RBCC enginesand the rigid body characteristics of the vehicle are shownin Fig. 9 with the intent to reveal the effect of the externalnozzle. Note that the performance of the total propulsion sys-tem including the external nozzle can be calculated as fol-lows:

Ttot ¼ T þ Text; ð10Þ

Isptot ¼ Isp 1þ Text

T

!: ð11Þ

In the early part of the flight (prior to 150 s), the externalnozzle has no effect because it is in an over-expansion con-dition. In reality, however, the nozzle can produce negativethrust in such a condition as experimentally shown by Isono

et al.26) After the altitude becomes sufficiently high and theexternal nozzle shifts to an under-expansion condition, onthe other hand, thrust and Isp are augmented.

Between different engine operating modes, there are largedifferences of pitching moments generated by RBCC enginesand the external nozzle. The requirement of maintainingpitching trim throughout multiple engine modes results inthe saturation of the elevon deflection angle even thoughthe vehicle has large elevons. If the propulsion system hasa gimbal or a thrust vectoring mechanism as in the case ofYokoyama et al.,9) this problem would be solved.

The longitudinal static stability of the airframe alone is lost(@MA=@� > 0) after 280 s or above Mach 4.3. In addition,since @Meng=@� > 0, the moment produced by the RBCC en-gines slightly deteriorates the stability. This is because, alarger angle of attack makes pre-compression prior to the in-let stronger, and as a result, larger thrust and larger positivepitching moment are produced by the RBCC engines. Theseinstabilities are compensated by the moment induced by theexternal nozzle. Stronger pre-compression increases the ex-haust pressure, and hence, the nozzle ramp pressure. Thisleads to larger negative pitching moment (i.e., @Mext=@� <

0). In total, positive stability is attained until 395 s and after445 s, and neutral stability is maintained otherwise.3.4. Flyability of the booster after staging

In the optimal solution, there is no remaining propellant inthe booster at the moment of staging. Therefore, the grossmass of the vehicle would increase if the flyback capabilityof the booster is imposed. To evaluate this increase in massquantitatively, trajectory optimization for the return flightmust be incorporated into the current MDO problem, and thismakes its size much larger. In this paper, the flyability of theobtained optimal booster shape after staging is evaluated interms of pitching trim and longitudinal static stability as a

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Fig. 9. Effect of the external nozzle on the propulsion performance and rigid body characteristics of the vehicle.


273©2017 JSASS

preliminary analysis before designing the return trajectory.The pitching moment characteristics of the booster alone

around the center of its dry mass are computed, and it wasrevealed that static stability cannot be retained above Mach6.8. The result at Mach 12.2, where the staging is conducted,is presented in Fig. 10(a). In order to remedy this instability,the installation of a body flap (10m width and 3m length) atthe fuselage base is examined. The effectiveness of the bodyflap is calculated using the hypersonic aerodynamic model inSection 2.3. The pitching moment curve at Mach 12.2 withthe deflection of the body flap (10 deg downward) is ob-tained as shown in Fig. 10(b), which means that the stabilityis achieved in a positive angle of attack. Summarizing theabove, the flyability of the booster after staging is confirmedin all speed ranges, while higher fidelity analysis is requiredin future work.

4. Conclusion

In this paper, a multidisciplinary conceptual design studyof a horizontal takeoff TSTO RLV with ethanol-fueledRBCC engines was conducted. An MDO framework com-posed of coupled analysis disciplines was built with the fol-lowing numerical models:� Estimation of the vehicle mass property was based on astatistical relationship.

� For the aerodynamic analysis, engineering-level CFDmethods were used, and surrogate modeling was applied.

� An analytical model of the interaction between theairframe and the propulsion system was developed, whichwas composed of pre-compression calculation, an RBCCengine model evaluation, and external nozzle flow compu-tation. This simplified methodology for evaluating theairframe-propulsion integration can be incorporated intoMDO studies of RLVs with ease, in contrast to costlyCFD simulations.

� In the flight trajectory computation based on point-massdynamics, pitching trim and longitudinal static stability

were also assessed while taking the influences of theRBCC engines into account.The vehicle design and the ascent flight trajectory of the

booster and those of the orbiter were simultaneously opti-mized in order to minimize the gross mass. Transportationof an 800 kg payload into a low Earth orbit was the assumedmission, and a vehicle with the gross mass of 581 t was ob-tained as the optimal design solution. The optimal flight tra-jectory has non-constant dynamic pressure in contrast to thetrajectory obtained in a previous study where airframe-pro-pulsion integration was not considered. The fuselage baseof the optimal booster shape is fully used as the external noz-zle, and it enhances propulsion performance. Additionally,the external nozzle enables longitudinal static stability tobe achieved throughout the ascent flight, whereas the vehicleairframe alone cannot retain stability above Mach 4.3.

In future work, the following improvements in the accu-racy of numerical models and in the problem formulationshould be considered for performing more reliable designstudies:� apply higher fidelity models to the estimation of vehiclemass and aerodynamic forces;

� employ a more accurate external nozzle flow model;� consider the return flight trajectories of the booster and theorbiter; and

� conduct thermal analysis.Since a gradient-based optimizer is employed in the MDO

methodology in this paper, one of its limitations is the lack ofthe search capability of the global solution. A possible strat-egy for solving this drawback is the application of a hybridoptimization algorithm, where a gradient-based optimizerand an evolutionary algorithm27) are responsible for trajec-tory design and vehicle design, respectively.

Acknowledgments

This study was partly supported by a collaborative research con-tract between the Japan Aerospace Exploration Agency and the Uni-versity of Tokyo.

References

1) Fujii, K., Ishimoto, S., Mugitani, T., and Minami, Y.: Present Statusand Prospects of JAXA’s Research on Future Space TransportationSystem, AIAA Paper 2012-5849, 2012.

2) Tani, K., Tomioka, S., Kato, K., Ueda, S., and Takegoshi, M.: RecentActivities in Research of the Combined Cycle Engine at JAXA, Trans.JSASS Aerospace Technology Japan, 8, ists27 (2010), pp. Ta_1–Ta_6.

3) Tomioka, S., Hiraiwa, T., Saito, T., Kato, K., Kodera, M., and Tani,K.: System Analysis of a Hydrocarbon-Fueled RBCC Engine Appliedto a TSTO Launch Vehicle, Trans. JSASS Aerospace Technology Ja-pan, 12, ists29 (2014), pp. Pa_91–Pa_99.

4) Kodera, M., Ogawa, H., Tomioka, S., and Ueda, S.: Multi-ObjectiveDesign and Trajectory Optimization of Space Transportation Systemswith RBCC Propulsion via Evolutionary Algorithms and Pseudospec-tral Methods, AIAA Paper 2014-629, 2014.

5) Ogawa, H., Kodera, M., Tomioka, S., and Ueda, S.: Multi-Phase Tra-jectory Optimization for Access-to-Space with RBCC-Powered TSTOvia Surrogate-Assisted Hybrid Evolutionary Algorithms IncorporatingPseudo-Spectral Methods, AIAA Paper 2014-2360, 2014.

6) Fujikawa, T., Tsuchiya, T., and Tomioka, S.: Multi-Objective, Multi-

−5 0 5 10 15−0.005

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0.005

0.01

Angle of attack [deg]

Elevon defl. = −10 [deg]Elevon defl. = 0 [deg]Elevon defl. = 10 [deg]C

M,c

g

(a) Clean configuration

−5 0 5 10 15 −0.01

−0.005

0

0.005

Angle of attack [deg]

Elevon defl. = −10 [deg]Elevon defl. = 0 [deg]Elevon defl. = 10 [deg]C

M,c

g

(b) Configuration with body flap deflection (10 deg downward)

Fig. 10. Pitching moment of the booster alone at Mach 12.2.


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disciplinary Design Optimization of TSTO Space Planes with RBCCEngines, AIAA Paper 2015-650, 2015.

7) Balling, R. J. andWilkinson, C. A.: Execution of Multidisciplinary De-sign Optimization Approaches on Common Test Problems, AIAA J.,35, 1 (1997), pp. 178–186.

8) Tsuchiya, T. and Mori, T.: Optimal Design of Two-Stage-to-OrbitSpace Planes with Airbreathing Engines, J. Spacecraft Rockets, 42,1 (2005), pp. 90–97.

9) Yokoyama, N., Suzuki, S., Tsuchiya, T., Taguchi, H., and Kanda, T.:Multidisciplinary Design Optimization of Space Plane ConsideringRigid Body Characteristics, J. Spacecraft Rockets, 44, 1 (2007),pp. 121–131.

10) Dufour, R., de Muelenaere, J., and Elham, A.: Trajectory Driven Mul-tidisciplinary Design Optimization of a Sub-Orbital Spaceplane UsingNon-Stationary Gaussian Process, Struct. Multidiscipl. Optim., 52, 4(2015), pp. 755–771.

11) Gill, P. E., Murray, W., and Saunders, M. A.: SNOPT: An SQP Algo-rithm for Large-Scale Constrained Optimization, SIAM Rev., 47, 1(2005), pp. 99–131.

12) Harloff, G. J. and Berkowitz, B. M.: HASA—Hypersonic AerospaceSizing Analysis for the Preliminary Design of Aerospace Vehicles,NASA CR-182226, 1988.

13) Magnus, A. E. and Epton, M. A.: PANAIR—AComputer Program forPredicting Subsonic or Supersonic Linear Potential Flows about Arbi-trary Configurations Using a Higher Order Panel Method, Vol. 1.Theory Document, NASA CR-3251, 1980.

14) Pittman, J. L.: Application of Supersonic Linear Theory and Hyperson-ic Impact Methods to Three Nonslender Hypersonic Airplane Con-cepts at Mach Numbers from 1.10 to 2.86, NASA TP-1539, 1979.

15) Bonner, E., Clever, W., and Dunn, K.: Aerodynamic Preliminary Anal-ysis System 2. Part 1: Theory, NASA CR-182076, 1991, p. 58.

16) White, F. M.: Viscous Fluid Flow, McGraw-Hill, New York, 1974,p. 639.

17) Forrester, A. I. J., Sóbester, A., and Keane, A. J.: Engineering Design

via Surrogate Modelling: A Practical Guide, John Wiley & Sons, Chi-chester, 2008.

18) Thompson, M. J.: A Note on the Calculation of Oblique Shock-WaveCharacteristics, J. Aeronaut. Sci., 17, 11 (1950), p. 744.

19) Beachkofski, B. K. and Grandhi, R. V.: Improved Distributed Hyper-cube Sampling, AIAA Paper 2002-1274, 2002.

20) NOAA, NASA, and U.S. Air Force: U.S. Standard Atmosphere, 1976,NASA TM-X-74335, 1976.

21) Raymer, D. P.: Aircraft Design: A Conceptual Approach, Fifth Edi-tion, AIAA, Washington, DC, 2012, pp. 688–690.

22) Patterson, M. A., Weinstein, M., and Rao, A. V.: An Efficient Over-loaded Method for Computing Derivatives of Mathematical Functionsin MATLAB, ACM Trans. Math. Softw., 39, 3 (2013), pp. 17:1–17:36.

23) Benson, D. A., Huntington, G. T., Thorvaldsen, T. P., and Rao, A. V.:Direct Trajectory Optimization and Costate Estimation via an Orthog-onal Collocation Method, J. Guid. Control Dynam., 29, 6 (2006),pp. 1435–1440.

24) Fujikawa, T. and Tsuchiya, T.: Enhanced Mesh Refinement in Numer-ical Optimal Control Using Pseudospectral Methods, SICE J. ControlMeas. Syst. Integr., 7, 3 (2014), pp. 159–167.

25) Taylor, J. W. R. (ed.): Jane’s All the World’s Aircraft 1989–90, Jane’sInformation Group, London, 1989, pp. 242–243.

26) Isono, T., Tomioka, S., Takahashi, H., Sakuranaka, N., Ono, M., andMikoshiba, R.: Evaluation of Influence of Flight Environment uponExternal Nozzle Flow in Wave-Rider Type Spaceplane, Proceedingsof the 58th Space Sciences and Technology Conference, JSASS-2014-4721, Nov. 2014 (in Japanese).

27) Zhou, A., Qu, B.-Y., Li, H., Zhao, S.-Z., Suganthan, P. N., and Zhang,Q.: Multiobjective Evolutionary Algorithms: A Survey of the State ofthe Art, Swarm Evol. Comput., 1, 1 (2011), pp. 32–49.

T. ShimadaAssociate Editor


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multidisciplinary design optimization of a two-stage-to

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