Original Article
Multidisciplinary design optimization of space transportation control system using genetic algorithm
Proc IMechE Part G: J Aerospace Engineering 2014, Vol. 228(4) 518–529 ! IMechE 2013 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954410013475573 uk.sagepub.com/jaero
Jafar Roshanian, Masoud Ebrahimi, Ehsan Taheri and Ali Asghar Bataleblu
Abstract In this study, due to the innate trans-atmospheric nature of flight of the space transportation system, assessment of the control discipline interaction with aerodynamic, weights and sizing, external fin-stabilizers configuration, and trajectory disciplines in an multidisciplinary design optimization-based platform has been addressed. Parameters considered for the control sub-system optimization are external stabilizing fins geometrical characteristics and attitude control vernier motors thrust value. Specifically, this article addresses optimization of fin–body combinations with geometric constraints for minimizing control moment required by vernier motors as well as total possible control sub-system weight satisfying design constraints. Results show that using external stabilizer fins is not economical from energetic stand point for space transportation system, but is necessary for control subsystems when there are deflection constraints for vernier motors. Keywords Multidisciplinary design optimization, space transportation system, genetic algorithm Date received: 28 March 2012; accepted: 3 December 2012
Introduction The design of large, complex, aerospace systems, such as space transportation system (STS), requires making appropriate compromises to achieve balance among many coupled objectives such as safety, high performance, simple operations, and low cost. The aforementioned factors lead the aerospace design engineers look for a method to consider objectives of higher priority in the design process while common interactions between most influential disciplines are still preserved. Therefore, multidisciplinary design optimization (MDO) techniques, that integrate the simulation tools of constituent disciplines and optimize the entire system, have offered a great advantage to the conceptual design of STS.1–3 However, traditionally, the vehicle design process is decoupled into smaller, more manageable, multidisciplinary problems4 that address vehicle performance. The two main categories are: (a) ascent problem, primarily involving trajectory, weights and sizing, and propulsion analyses and (b) entry problem, emphasizing geometry, aerodynamics, trajectory, heating, structures, and controls. For example in Barnum et al.,5 application of system sensitivity analysis method6 to the
optimization of a three surface civil transport aircraft with constraints on the aircraft range, number of passengers, stability requirement, and maximum wingspan was investigated. As another example, application of MDO of Subsonic Fixed-Wing Unmanned Aerial was done in Gundlach.7 The design variables were wing span, aspect ratio, taper ratio, and thickness to chord ratio. Also in Giunta et al.,8 MDO of advanced aircraft configuration is done and the coordinates of a cranked-wing are design variables. Although, there exists many works in wing design optimization in the field of aircraft and unmanned aerial vehicle design. In the field of STSs, wing design optimization has been mostly done on reusable launch vehicles (RLVs). The wing in RLV provides a great deal of design flexibility while dealing with issues of cross range, entry heating, flight control, and landing speed.9,10 In Yokoyama et al.,11 MDO Lab, Faculty of Aerospace Engineering, K.N. Toosi University of Technology, Tehran, Iran Corresponding author: Masoud Ebrahimi, MDO Lab, Faculty of Aerospace Engineering, K.N. Toosi University of Technology, East Vafadar Boulevard, 4th Tehranpars Square, 16569-83911, Tehran, Iran. Email:
[email protected]
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MDO of a space plane configuration optimization considering trimming and stability constraints has been investigated. Also, in Jean-Marius,12 controllability has been considered to allow better tackle the lateral effort structural sizing aspects. In Lesieutre et al.,13 MDO of missile configurations and fin planforms minimizing fin hinge moment has been investigated and precise aerodynamic methods covering subsonic and supersonic ranges have been considered in optimization process. In Yokoyama et al.,14 trim and the stability of a single-stage-to-orbit is considered. However, trans-atmospheric flight such as the ascent of a STS includes a considerable migration in both the aerodynamic center and the center of gravity (CG). The migrations can cause severe instability and produce large moments that must be counteracted by control subsystem. Furthermore, the integration design of stabilizer fins (SFs) into the airframe, which is important to improve the stability performance, substantially affects the control subsystem through aerodynamic moment arm. Although static stability is desirable, it is not required for modern flight control systems and, in fact, neutral stability may be better from the standpoint of control power requirements. However, in the past MDO studies of STS, the simulation model that covers both these control subsystem and the stabilizer-fin-airframe integration has rarely been considered in detail. Therefore, this study formulates and solves an MDO problem considering the stabilizer–airframe integration and its effect on the control subsystem in terms of required control moment. Also, it provides the capability to design SFs with respect to (wrt) the designer demands in terms of maximum deflection of verniers and available vernier level of thrust. The sections that follow first give an overview of the base vehicle and design problem, then MDO framework implemented in this study is introduced, afterwards principal disciplines involved and those areas of greatest needs for modeling will be explained. Next, a brief history is given of the vehicle synthesis and optimization frameworks within which these disciplines have been incorporated. Finally, a discussion of optimization methods and results will be presented.
(LEO) and can have separation components necessary for orbital flight described as upper stage. The coupling among disciplines in the conceptual design of a two-stage Small Solid Propellant Expendable Launch Vehicle (SSPELV) was investigated in Jodei et al.15 which is the base vehicle considered in this study. There, the problem was minimizing the weight of an SSPELV to launch a satellite with a known mass and inject it into a given LEO. Figure 1 shows the basic airframe configuration of the vehicle. In this section, the proposed algorithm for considering disciplines interactions is demonstrated. There have been many different attitude control mechanisms in STSs design. Of the most used are vernier motors (VMs) and jet vanes. The latter one is an old method reducing the effective thrust and is to some extant obsolete. On the other hand, external SFs have a considerable effect on the point of center of pressure (CP) and consequently on the produced aerodynamic forces and moments through the static stability margin. In terms of subsystem design, the SFs determine the vernier thruster actuator sizing along with the level of the thrust of the vernier motor. The actuators influence the vehicle weight directly through their size and power requirements. Through the interaction of these two devices, a proper compromise between the required force of VMs and their mass from one hand, and the geometrical characteristics of the stabilizers and their corresponding mass on the other hand can be achieved. In the following section, selection of the design parameters, objective functions, and constraints will be explained. Figure 2 shows the corresponding plan view of the SFs design variables to be optimized. Parameters selected for the stabilizer geometrical definition are chord of the root (Cr ), semi-wet span (Spanwet), and leading edge sweep angle (). Also the thrust of each vernier motor is the other design parameter to consider mass variation models in the design and provide a trade-off study. So the general
Design problem and methodology Design problem While current investments in STS design and technology are intended to enable a viable, RLV business, such a business does not yet exist, and the future production will continue to be small, on the order of tens rather than hundreds, for the next decade or two. However, unlike single-stage vehicles, the feasibility of the two-stage-to-orbit (TSTO) is not severely limited because it must not be put on the low-earth orbit
Figure 1. Basic configuration.
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design vector is X ¼ Tc,1 , Cr , Spanwet , . Current and future STSs are being designed for internal carriage. Internal carriage sets limit on fin span due to stowage requirements and should be considered. The fin spans due to work on STS stabilization have positive effects on control system. On the other hand, fin spans absence can lead to taking less space. Therefore, the vehicle transportation and stowage became easier. However, the compromising between these subjects (i.e. fin spans absence or existence) is the main goal of this study.
Design methodology Selection of the objective function, the design variables, and constraints will have a considerable effect on the final design. Many objective functions can be considered for this problem. Minimum LV weight is used by the design community as one of the most important objective functions. However, it is evident that adding SFs mass to the base vehicle will increase the total mass and drag inevitably. It maybe thought that using stabilizers, less deflection is required and most of the VMs thrust will add up to the axial force and compensate the mass increase, but the vehicle drag increases as the vehicle speeds up and the total axial acceleration will decrease as the results show. Consequently, to expect a decrease in total mass is totally wrong except that there exists an accurate model for actuators mass to count for control subsystem mass reduction due to less control moment. The other objective function that can be considered is the total required control moment. If the objective function is considered as integral of the absolute value of the total control moment in pitch plane during the first stage t¼t Z s1
J ¼ min t¼0 t¼t Z s1
¼ min
Myc dt 2Tc,1 sinð2 Þ xT xc:g dt c
t¼0
ð1Þ
in which Tc,1 is the thrust of a vernier motor, 2 the deflection of the vernier nozzle, and xTc xc:g the instantaneous control moment arm. It is clear that vernier thrust has the greatest contribution. In addition, exertion of a constraint on the minimum limit of vernier motor deflection is meaningless. Hence, minimization of the objective function will follow a direction that omits the VMs or as another explanation in case of a lower and upper bound for vernier thrust, it will result in the lower bound. To eliminate the aforementioned problem, the following objective function with the total lift-off mass and constraint on the maximum allowed vernier deflection is proposed J ¼ min fm0 g ¼ min fmPTC þ mVerniers þ mstabilizer g Constraint : 2, max atmospheric flight 42 allowed ð2Þ in which mPTC is the predetermined test case total mass that did not have control system (i.e. VMs and SFs), mVerniers the total mass of the VMs including propellant and nozzles, and mstabilizer the total mass of the stabilizer set. In this study, estimation of a control system mass of a predetermined test case introduced in Jodei et al.15 is considered. The characteristics of predetermined test case are presented in Table 1.15 Here, the effect of control system mass is under survey and it is assumed that mPTC (the predetermined test case total mass that did not have control system) is constant for all configurations, therefore, VMs and SFs have not been included in mPTC . In the constraint equation, j2, max atmospheric flight j is the absolute maximum deflection angle obtained through the atmospheric flight and 2 allowed is the authorized upper limit for vernier deflection angle due to technology or actuator power limits. Selection of the above objective function is accordance with the fact that the lower total mass at lift-off Table 1. Representative values of the predetermined test case parameters. Parameters
Figure 2. Shape and vernier thrust design variables.
Optimum value
Propellant mass (kg) Average thrust (kN) Burning time (s) Length (m) Diameter (m) System mass (kg)
First stage motor
36,103 1599 63.8 6 2.5 4401
Propellant mass (kg) Average thrust (kN) Burning time (s) Length (m) Diameter (m) Structure mass (kg)
Second stage motor
8693 205 128.6 5.4 1.3 884
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with the same performance is equal to more useful payload. Exercising the above objective function will lead to the optimization and is explained accordingly. Two procedures for reduction of deflection angle are conceived. The first one is the growth in VMs thrust and the second one is the enlargement of the stabilizers area. The first strategy will lead to an increase in mVemiers while the second one also results in mstabilizer increase. So as a consequence, if the SFs are omitted, the whole control moment should be provided with VMs through the increase at both Tc,1 and 2 terms. Since 2 is limited, a greater vernier thrust is needed and finally the control subsystem mass will increase in the same manner. On the other hand, if a big stabilizer is used, it has two drawbacks, the first one is its mass and drag increase and the second one is making the STS much more statically stable and it again needs more control moment. Furthermore, structural limitations should be considered. It should be noted that SFs show their influence in static stability term that produces the aerodynamic moment. As it was explained, it is sought to have neutral static stability ðxCP xCG Þ during the atmospheric phase of flight. Such trade-off study is impossible without consideration of the control discipline that results in vernier deflection 2 to track a pre-determined command in terms of commanded pitch rate or attitude.
MDO framework While there are several advanced MDO methods, this study adopts all-at-once (AAO). In AAO method, all
the nonlinear equations in the MDO problem are simultaneously solved without the inner iteration of each analysis program. The advantage of AAO method is that the robustness and the rate of convergence are generally better than those of the other approaches. However, it is difficult to embed complicated analysis programs to the framework of AAO method. In this problem, the design variables are few and this method is a good candidate. As shown in Figure 3, the MDO framework consists of an analysis process and an optimization process. Also, the design variable flow is depicted. It is desirable that the analysis model of each discipline is sufficiently accurate and reliable to obtain practical results by MDO for conceptual design. In the following sections, each discipline is described.
Analysis modules Weight and sizing The discipline that is needed for the conceptual design and synthesis of a STS is mass properties analysis. For this project, the authors have taken a simple approach to integrate mass properties analysis into the design environment. According to a predetermined test case, the general mass characteristics of the vehicle are specified except for the stabilizer and VMs thrust. Once the input design and geometric parameters are determined, a set of mass estimating relationships (MERs) is used to determine the initial estimate of the mass properties of the vehicle.15 For this project, those
Figure 3. Schematic view of MDO framework. MDO: multidisciplinary design optimization.
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MERs that are based on component geometry are used. An example of a geometry-based MER is equation (3). This equation estimates the weight of the vertical tail based on the surface area of the tail. mstablizers ¼ 5 S1:05 stablizers
ð3Þ
which Sstablizers is planform area of the four stabilizer surfaces. Also the mass of the vernier thrusters was calculated using equation (4). mvernier
propellants
¼4
Tc,1 tb1 Isp g
ð4Þ
in which Isp is the specific impulse of the propellant and tb1 the burn time of the first stage. On the basis of weights and locations of components, the position of the vehicle’s CG, mass and inertial tensor at any flight condition is calculated. The main solid propellant motor burning and its moment of inertia (MOI) calculation are done according to DeGrafft.16
Aerodynamic forces and moments This study involves estimating the STS aerodynamic properties in different flow field regions during the atmospheric flight, which ranges from subsonic to hypersonic speeds. Due to the accuracy needed in the consideration of external configuration aerodynamic model, a more detailed calculation of static and dynamic stability derivatives coefficients is considered. The formatting of force coefficients is mostly in body coordinates with the positive sense following the direction of the body axes17 B CX fa ¼ qS
CY
CZ
ð5Þ
where q is the dynamic pressure and S the reference area. The format of the moments, referenced to the body center of mass (CM) has always been in body coordinates ½m a B ¼ qSL Cl
Cm
Cn
ð6Þ
with term L as the reference length. Aerodynamic coefficients that are used in this program are calculated as the following17 C CX q 2:V q b CYr r þ CYp p CY ¼ CY0 þ 2:V C CZ q CZ ¼ CZ0 þ 2:V q b Clr r þ Clp p Cl ¼ Cl0 þ 2:V C Cz Cmq q þ XcgR Xcg Cm ¼ Cm0 þ 2:V C CY b Cnr r þ Cnp p Cn ¼ Cn0 XcgR Xcg þ 2:V b
For this effort, the team was comfortable with using Missile DATCOM18 to determine the conceptual level aerodynamic characteristics of the vehicle. This program is one of the most widely used programs for conceptual design and its validity is investigated for diverse configurations at many angles of attack and Mach numbers.19–21 It is capable of quickly and economically estimating the aerodynamics of wide variety of configuration designs and it has the predictive accuracy suitable for conceptual and preliminary designs especially in the field of MDO. Application of a more detailed aerodynamic coefficients relation is investigated in STS six-degree-of-freedom (6DoF) in Taheri et al.22 The coefficients are calculated based on the combinations of representative values of the flight Altitude h, Mach number M, the angle of attack (AOA) a, and the sideslip angle , as presented in Table 2. Figure 4 illustrates CX, CY, CZ, and Cm versus AOA and Mach number. Given a shape of the vehicle based on the design variables, the aerodynamic coefficients at the representative values are calculated based on a four-dimensional linear interpolation of the aerodynamic data tables.
Control algorithm Pitch program of a STS is provided by guidance system. Some guidance systems provide only a pitch program whereas some others also require that the control system be capable of accepting body commanded rates. The control algorithm used in this study is based on control channels decoupling and gain scheduling. These methods are presented in detail in Roshanian et al.23 Therefore, in order to avoid increasing the materials of the manuscript drastically, we do not mention them here. The combination of simple gain scheduling with decoupling, based on the developed design method, can lead to better adaptation characteristics than with simple gain scheduling. This algorithm provides the VMs deflection angle through which the control moment and forces can be calculated. Gains of the control law are generally selected by automatic control theory techniques, such as pole-zero placement method of the closed-loop transfer function.
CX ¼ CX0 þ
Table 2. Representative values of the flight condition parameters for aerodynamic analysis.
ð7Þ
Parameters
Representative values
h (m) ( ) M ( )
[2000.0,20000.0,40000.0,70000.0] [3.0,3.0] [0.05,0.3,0.8,1.3,2.0,3.0,4.0,5.0,7.0,9.0] [6.0,4.0,2.0,0.0,2.0]
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Trajectory The final discipline needed to ‘close’ the design of a STS is trajectory simulation. Trajectory simulation is used in an iterative process to determine the final, ‘closed’, conceptual design. In this article, the 6DoFmodel of the flight is analyzed.17 The characteristics of the simulation are rotating, elliptical Earth model. Environment routines use Standard atmosphere model; these models are the most accurate and perfect models that can be used for conceptual design. In the trajectory code, the equations of motion are numerically integrated (fourth order Runge– Kutta) from initial state conditions. Equations of motion can be divided into two major parts: translational and rotational equations. The translational equations express inertial acceleration of the instantaneous CM of the STS as a function of applied forces while the rotational equations calculate angular rates
using exerted moments. Newton’s second law governs the translational DOFs and Euler’s law controls the attitude dynamics. Both must be referenced to an inertial reference frame, which includes not just the linear and angular moments but also their time derivatives. Using Newton’s second and Euler laws, the following results are obtained17 Translational equations
I B GI dvIB 1 BI ¼ T fa,p,c þ T ½ gG m dt
ð8Þ
Rotational equations BI B B 1 d! ¼ IBB dt B B B B
BI IBB !BI þ ma,p,c
ð9Þ
Figure 4. Aerodynamic coefficients versus AOA and Mach number: (a) CX versus AOA and Mach number, (b) CY versus AOA and Mach number, (c) CZ versus AOA and Mach number, and (d) Cm versus AOA and Mach number. AOA: angle of attack.
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The index B indicates the body coordinates. For example, the notation IBC reflects the MOI tensor of body (frame) B referred to the reference point C. Therefore, the notation IBB reflects the MOI tensor of body referred to the body CM. The ½ gG term in translational equations presents the gravity acceleration B in geocentric coordinate system (CS). The fa,p,c term presents the resultant aerodynamic, propulsion and control forces in the body CS, respectively. As they are calculated in the body CS, coordinate transformation matrix ½TIB is used B to express them in the inertial CS. The term IBB is the inertial of the vehicle in the body tensor B frame. The ma,p,c term is summation of aerodynamics, propulsive and control moments in body CS. The first equation results the vehicle velocity and position while the second one gives the body rates. In the following sections, calculation of forces and moments is explained.
Search method
Results and discussion Using the AAO-based MDO approach, the design of the vehicle and its optimization were studied in four cases: (a) base vehicle (without SFs); (b) with static-margin lower limit set to 0.1D (with SFs), which D is the vehicle’s first-stage diameter; (c) optimized with design vector as X ¼ Cr , Spanwet , ; (d) optimized with vector as design X ¼ Tc,1 , Cr , Spanwet , . The cases a and b were presented for comparison and trade-off study as the two extreme cases. For the proposed method using specific ranges of design variables, search space is limited to those presented in Table 4. hallowed and Vallowed are considered as penalty due to the decrease in vernier thrust value. The effect of fin-stabilizers on the performance of the STS has been considered especially those concerning the control subsystem. Table 5 presents the specifications of the four cases. Figure 5 shows the corresponding tail configurations with 20 and 45 off-set (for clarity and comparison) for cases 3 and 4. Cases
One of the best-known stochastic search methods is genetic algorithm (GA), which is designed to mimic evolutionaryselection.24 In recent years, some researchers have done successful works applying GA algorithms within MDO frameworks to solve design problems both in control and configuration disciplines.25,26 In this study, GA is selected as the optimizer. Before specifying GA parameters, an extensive parametric study was conducted for the problem in hand by varying one design parameter at a time. Selecting an optimal combination of GA parameters is very difficult because each of the GA parameters is varied individually and the number of combinations of the GA parameters is infinite. Parameters of the GA algorithm used in this research are given in Table 3. A population size of 100 is used. The size of the population must be related to the size of the search space, insuring a sufficient number of points for the evolutionary algorithm prospect. A tournament selection was used to allow for minimization and to avoid potential scaling concerns; both the population size and the string length influence the choice of the mutation probability, so this also varied with each problem. The maximum population iteration was used as a stop criterion, i.e. the search algorithm stops after 50 iterations.
Table 4. Constraints imposed on the shape design parameters. 0:24Cr 43 (m) 2, max atmospheric flight 435 (cases a and b)
wet 0.1 4 Spanwet 490 tan1 Span Cr 4 1 (m) hmax ,ref hmax 4hallowed 10 4490 ( ) 20, 0004Tc,1 422, 000 (N) Vmax ,ref Vmax 4Vallowed 2, max atmospheric flight 435 (cases c and d)
Table 3. Parameters setting for GA.
Total mass (kg) 52,896.4 53,227.9 52,979.6 52,843.36 21,500 21,500 21,500 20,125 Tc,1 (N) R t¼ts1 M dt (N.m.s) 2,815,328.9 1,539,202.1 1,227,222.8 1,228,901 yc t¼0
Population size Iteration Mutation Crossover probability Selection strategy GA: genetic algorithm.
100 50 Gaussian 0.8 Tournament
Table 5. Specifications of the four cases. Parameters
Case 1
Case 2
Case 3
Case 4
Gross mass (kg)
50,754
50,754
50,754
50,754
mstabilizers (kg)
–
331.8
83.56
75.59
mVerniers (kg)
2085.4
2085.4
2085.4
1957.12
2 allowed ( )
35
35
35
35
Cr (m)
–
2.3
1.054
1.254
Spanwet (m)
–
1.02
0.83
0.537
( )
–
10
34.7
13.6
hmax (m)
66,393.766 65,443.47
66,201.4
66,107.34
Vmax (m/s)
2923.91
2912.2
2908.02
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Figure 5. 2D and 3D shapes of the stabilizer cases: (a) STS configuration, (b) 3D graph of fins, and (c) 2D graph of fins. STS: space transportation system.
(a)
(b)
10 9
50
7 6
Height (Km)
Height *10 (Km) & Mach
Case a Case b Case c Case d
60
Mach Height
8
70
5 4 3
40
30
20
2 10
1 0
0
10
20
30
40
50
60
0
70
0
1
2
3
t (sec)
(c)
Case a Case b Case c Case d Alpha
0.5
–0.5 –1 –1.5 –2
5 Mach
6
7
8
1
9
10
Case a Case b Case c Case d
0.5 (Xcp –Xcg )/D & α (Deg)
(Xcp –Xcg )/D & α(Deg)
0
(d)
4
α
0 –0.5 –1 –1.5 –2 –2.5
–2.5 –3 1
2
3
4
5 Mach
6
7
8
9
10
20
30
40
50
60
t (sec)
Figure 6. Flight parameters history of: (a) height and Mach number wrt flight time for case a, (b) height wrt Mach number, (c) static margin and AOA wrt Mach, and (d) history of static margin and AOA wrt time. AOA: angle of attack; wrt: with regard to.
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4
(a)
x 10
0
(b)
–2 –4
Case a Case b Case c Case d
20
10 –6
d 2 (Deg)
Axial Aerodynamic Force (N)
30
Case a Case b Case c Case d
–8 –10 –12
0
–10
–20
–14 –30 –16 –18
0
10
20
30
40
50
60
–40 0
70
10
20
5
(c)
2
x 10
(d) Case a Case b Case c Case d
1.5
50
60
70
8.6
x 10
4
Case a Case b Case c Case d
8.4 Axial Control Force (N)
Myc (N.m)
40
8.5
1
0.5
0
8.3
8.2
–0.5
8.1
–1
8
–1.5
30 t (sec)
t (sec)
0
10
20
40
30
50
60
70
7.9
0
10
20
t (sec)
40
30
50
60
70
t (sec)
Figure 7. Flight parameters history of: (a) axial aerodynamic force wrt time, (b) deflection angle wrt time, (c) control moment wrt time and (d) axial control force wrt time. wrt: with regard to.
2–4 are colored in yellow, blue, and green, respectively. Figure 6(a) shows the history of height and Mach number wrt the flight time for test case a. It is obvious that the vehicles fly in all the regimes including subsonic, supersonic, and hypersonic. The STS becomes supersonic after 13 s of flight. Also, it spends approximately 40 s at the denser parts of the atmosphere between altitudes of 2–45 km. Figure 6(b) shows the history of height wrt the flight Mach number for all cases. Figure 6(c) shows the history of the static margin wrt the flight Mach number. It illustrates the hypersonic and supersonic effects of integrating a SF with the two-segment circular body designed to provide efficient ascent performance. The STS flights vertically 10 s and starts rotating afterwards in accordance with the provided commanded pitch rate to have a small AOA at maximum dynamic pressure. For case a, the static margin shows decreases as it reaches Mach 2, then it has an approximately constant value until it passes Mach 5
which is equal to height of 30 km and the aerodynamic effects become negligible. It should be noted that the static margin is big and should be counteracted with the VMs. While the reverse is true for case b, which is statically stable, better results are provided in cases c and d. Supersonically, the stabilizers provide a much better CP match with the vehicle CG. Figure 6(d) shows the history of static margin and AOA variations wrt time. As it is evident from Figure 7(a), the most influential part of flight in terms of axial aerodynamic force is in the height range of 2–40 km. Also, case a has the lowest drag because it employs no fin and case b has the highest drag due to its big SFs. Figure 7(b) shows the history of the deflection angle wrt the flight time in all the cases. Cases a and b are the most demanding cases in terms of vernier motor deflection angle as the two extreme cases. It is interesting to note that in cases c and d, the deflection has had an approximately constant value which also makes the control moment to some extant fixed during the atmospheric phase, as
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(b)
(a)
Case a Case b Case c Case d
0
40
38
–1 –1.5 –2 –2.5
32
2
5
10
15
20 t (sec)
25
30
–3 0
35
x 105
20
30 40 Height (Km)
50
60
70
0.5
0
d (Deg)
10
0
–10
–0.5
–20
–1
–30
–2.5
–2
–1.5
–1
–0.5
0
0.5
Case a Case b Case c Case d
20
1
–1.5 –3
10
(d) 30 Case a Case b Case c Case d
1.5
Myc (N.m)
–0.5
36
34
(c)
Case a Case b Case c Case d
0.5
(Xcp–X cg)/D
Axial Acceleration (m/s 2)
42
1
1
–40 –3
–2.5
–2
–1.5
–1
–0.5
0
0.5
1
(X cp –X cg )/D
(X cp –X cg )/D
Figure 8. Flight parameters history of: (a) axial acceleration wrt time, (b) static margin wrt height, (c) control moment wrt static margin, and (d) deflection wrt static margin. wrt: with regard to.
depicted in Figure 7(c). From Figure 7(d), in test case a, the axial control force is diminished due to the high deflection of VMs. In case c, this decrease has been relieved by employing SFs. Figure 8(a) clearly explains that case a, has the highest axial acceleration, hence resulted in the highest velocity and attitude mentioned in Table 5 for a specified command pitch rate. Cases c and d show a similar trend but case b is the worst one with the biggest SFs. Figure 8(b) shows the variation of static margin wrt height. Figure 8(c) provides a good graph for analyzing the results. It clearly shows the range of control moment variation in terms of static margin change. Case a shows the change of control moment in a vast change of static margin while case b shows the same magnitude of control moment in a limited range of static margin variation. Also, cases c and d show limited change of control moment in a limited range of static margin change. The same manner is shown in Figure 8(d) for deflection angle.
Conclusion The design of TSTO vehicle is a particularly challenging design problem, requiring special approaches to solve unique design issues and achieve a concept which successfully integrates the requirements. Because of the importance of the control subsystem, the design strategy must feature a highly efficient control system. The fin-stabilizer concept along with vernier thrusters provides an approach to solving this challenging problem and provides the ability to make the required trade-offs. To this end, an MDO problem of a TSTO space plane was formulated and solved in this study. The analysis model covered the rigid body characteristics such as the control as well as the airframe–SFs integration design. Using the AAObased MDO approach, the design of the vehicle was considered in four cases and successfully optimized. The trade-offs between the performance and the airframe geometry favorable for the rigid body characteristics (i.e. the control) were observed. The only way
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we can reach lower total mass is by utilizing lighter VMs. However, if we reduce the level of the thrust of VMs, the deflection angle will increase to provide the same amount of control moment of case a. In case d, the simultaneous effects of changing VMs and sizing SFs are studied to see if a better configuration is available or not. Case d is the best solution we obtained by applying GA to the defined objective function, i.e. equation (2) subjected to the ranges and constraint mentioned in Table 3. The results show that for the considered range of thrust level for VMs, the algorithm converged to a configuration which is 53 kg lighter than the base vehicle. However, the difference between the final height and final velocity of cases a and d seems to be more important. In fact, the effect of aerodynamic drag due to the added SFs surpasses the aggregate advantage of having lighter VMs with lower thrust level and lower deflection angles in the whole atmospheric phase of flight. However, the promising result is that with the SFs integrated into the body frame, less deflection is required and the energetic difference between the two configurations is not considerable. It become clear that using external SFs is not economical from energetic standpoint for STS, but is necessary for control subsystems when there are deflection constraints for VMs. Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
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