Small Solid Propellant Launch Vehicle Mixed Design Optimization Approach

ABSTRACT: For a small country with limited research budget and lack of advanced space technology, it is imperative to find new approaches for the development of low-cost launch vehicles (LV), which is, among all possibilities, an interesting option for rapid access to space, focused on integration of acquired components complemented with indigenously developed subsystems. This approach requires the cooperation of developed countries with huge experience and knowledge in LV development and operations. The main objective is to develop a small three stage solid propellant LV capable of delivering a payload of 100 kg to a circular low earth orbit of 600 km altitude, with the first and second stage solid rocket motors (SRM) hypothetically acquired from different countries and the third one designed and produced domestically in accordance with the production and technological capability. This approach provides main advantages such as: reduction in total time to access the space and to master the basic knowledge of launch operations. For this purpose, an integer continuous genetic algorithm global optimization method was selected and implemented, the SRM characteristics of the first and second stage were considered as integer variables, whereas the design variables of the third stage SRM and the trajectory variable were considered as continuous. A multi discipline feasible (MDF) framework was implemented along with the propulsion, aerodynamic, mass and trajectory models. Despite their particular characteristics and constraints, the results show highly acceptable values, and the approach proved to be reliable for conceptual design level.


INTRODUCTION
The last decade may be characterized by an increased number of small satellites delivered into the low earth orbit (LEO), and this tendency will be dominant in the coming years.
Small satellites have a reduced manufacturing cost, and are relatively easy to operate and maintain.Furthermore, the miniaturization of technology makes possible its delivery into space by using small cost effective launch vehicles (LV).
Small countries generally have a limited research budget oriented to space technology development, however, nowadays it is possible to deliver a small satellite into orbit with a reasonable budget, considering the cooperation with technologically more advanced countries.
This research was focused on finding a way to have rapid access into space and to master the basic knowledge of space development and operations.In such a way, several options had been analyzed, among them the most suitable alternative in terms of economic investment and development time resulted in a small solid propellant LV with mixed design configuration, involving a strong cooperation with different countries.
The strategy considered here prioritizes the technology integration over expensive and time consuming new development, this means that complex and advanced devices were acquired and complemented with indigenous manufactured devices using available resources and technology.
As a result, a three stage solid propellant LV was configured, where the first and second stage solid rocket motors (SRM) were acquired from different providers, complemented with a locally developed third stage SRM, which was designed and optimized to accomplish the specific mission.Villanueva, F.M., Linshu, H. and Dajun, X.In our research, a mixed integer continuous variables genetic algorithm (GA) method has been used in order to optimize the overall confi guration of the LV.

LAUnCh VEhiCLE DEFiniTiOn
A small three stage solid propellant LV in tandem confi guration is considered for this research.Th e mission is to deliver a 100 kg payload to a circular LEO of 600 km of altitude.Th e payload volume requirements and the instrument module weight were specifi ed beforehand in mission defi nition analysis and are shown in Table 1.

COnSiDERED SOLiD ROCKET mOTORS
Th e considered SRM are listed in the Table 2 and were selected based on the variety of design characteristics.However, it is possible to add additional parameters such as cost, availability, technology complexity, country of origin among others.

pROpULSiOn AnALYSiS
Th e propulsion analysis has been conducted for all three stages of the LV, using the classical approach presented in Sutton and Biblarz (2001) and He (2004a;2004b).For the third stage SRM, a detailed analysis was conducted, considering the properties of the propellant.In this analysis, the burning surface is considered constant by introducing a grain geometry shape coeffi cient, k s , the burning surface of the grain S b can be calculated as: where, L m is the rocket motor cylindrical length and D m the diameter.
Th e burning time t b , grain mass m gn , and mass fl ow rate m gn of the grain are calculated as: (2)

Variables
Units Value

Payload kg 100
Fairing mass kg 50 Instrument module kg 50 Payload deployment module kg 50 (3) (4) (5) where, u is the burning rate of propellant, ρ gn density of the grain, L gn = L m + 0.314D m length of the grain, D gn = D m diameter of the grain, λ gn fi neness ratio of the grain (grain length/diameter), and η v the grain volumetric loading fraction.Th e expansion ratio ε, nozzle throat area A t , and nozzle exit area A e are calculated as: where, P c is the chamber pressure, p e exit pressure, R c = 296 J/(kg.K) gas constant, T c = 3300 K temperature in the combustion chamber, P c max = 1.1P c maximum value of chamber pressure, and k = 1.2 the specifi c heat ratio of gas.
Th e specifi c impulse I sp , and the thrust T can be calculated as: where, p a is the atmospheric pressure, I a sp average specific impulse, g acceleration due to gravity, and A e the nozzle exit area.

mASS AnALYSiS
Th e mass analysis was conducted for the entire LV, and is represented by the following equations:  He (2004a;2004b) provided a methodology and a detailed calculation of the third stage SRM structural mass.Th is design consisted in a classical metallic case made of high strength steel, ethylene propylene diene monomer (EPDM) for chamber insulation, and carbon phenolic for the nozzle.

AERODYnAmiC AnALYSiS
Th e aerodynamic coeffi cients were estimated using the Missile DATCOM 1997 digital (Blake, 1998).Th is soft ware is easy to use and implemented, and accurate enough for the conceptual design phase.Qazi and He (2005) and Villanueva et al. (2013) applied DATCOM in LV aerodynamics analysis.Th e lift and drag forces were calculated using the following relations: where, q is the dynamic pressure, D drag force, L lift force, S ref vehicle reference area, C L lift coeffi cient, and C D the drag coeffi cient.
Th e aerodynamic coeffi cients were calculated repeatedly for each LV confi guration, the selected Mach ranged from 0 to 8 and the angle of attack from −8 to +1 degrees.

TRAJECTORY AnALYSiS
Th e trajectory analysis considers a 3 degree of freedom (3DOF) model, which has been modeled in SIMULINK (Zipfel, 2007;Fleeman, 2001).Th e previously calculated aerodynamic coe ffi cients, the mass and the propulsion are the input parameters.In order to obtain a quick result, a 2D coordinate system was adopted, the LV fl ies as a point mass in a non rotating earth model.Figure 1 illustrates the forces acting on a LV and below a set of governing equations of motion (Xiao, 2001).Th e LV is fl ying in an inertial reference coordinate system XOY, with its origin located in the center of the earth.Furthermore, all forces applied to the LV were considered in relation to the body centered velocity coordinate systems xoy as shown in Fig. 1. (17) where, V is the velocity, m vehicle mass, θ pitch angle, η t rajectory angle, γ flight path angle, φ range angle, h height above ground, α angle of attack, and α prog (t) is the programmed angle of attack.
The axial and normal overload coefficients ensure the integrity of the LV in all phases of fl ight, and were calculated in a body centered velocity coordinate systems (xoy), as follows: The thrust to weight ratio gives an importan t value to evaluate the lift off characteristics of the LV: (20) Th e density variation with altitude can be calculated as: ( 21) Th e gravity varies with altitude and can be represented as: where, ρ 0 is the sea level density, R e radius of earth, β density scale height, and μ the earth gravitational parameter.
Th e mission requires to deliver the payload to an altitude h f with a circular orbital insertion velocity V f : (23)

TRAJECTORY phASES
Th e trajectory of the LV can be described as a composition of several phases, as presented by He (2004a), Qazi and He (2005) and Villanueva et al. (2013).For the present research, the trajectory was sectioned in seven phases, as shown in Fig. 2. Each phase has a specifi c fl ight characteristic as described next: • Vertical launch phase: Th is phase starts from the time of ignition of the fi rst stage SRM until the end of vertical fl ight time t v (t v = t 1 in Fig. 3), during this time the LV fl ies vertically with a fl ight path angle equal to 90 degrees.• Pitch over phase: During this phase, the LV maneuver with a negative angle of attack until the transonic velocity is reached.In this point, the angle of attack should approaches zero degrees.

FLiGhT pROGRAm FORmULATiOn
Th e fl ight profi le defi nes the performance and loads acting on the LV Consequently, its selection should be integrated in the optimization process.Figure 3 explains the variation of the angle of attack during the pitch over phase (He, 2004a;Xiao, 2001): where, α max is the maximum angle of attack, a m launch maneuver variable, t a time corresponding to maximum angle of attack, t time of fl ight, and t 1 the time of start of pitch over phase, coincident in value with time t v .

OBJECTi VE FUnCTiOn
Th ere can be diff erent objective functions for LV optimization problem, such as minimization of the LV cost, which can be obtained knowing the cost of the fi rst and second stage SRMs and the development cost of the third stage, and also the minimization of the development time, knowing the availability of the fi rst and second stage SRM and the development time of the third stage SRM.However, this analysis considers the minimization of the gross launch mass (m LV ).Th e mathematical description of design objective is as follows: where, g j is the inequality constraints, h k the equality constraints, X the set of variables, X lb the lower bound of variables and X ub the upper bound of variables.

DESiGn VARiABLES
The design variables are composed from integer (first and second stage SRMs), and continuous third s tage SRM and trajectory variables.Th ey are listed in Table 3 and can be represented as:

DESiGn COnSTRAinTS
The selections of constraints were oriented in order to satisfy the mission, to prevent any failure during fl ight, and to consider the limitation of the third stage manufacturing technology.Th ey are listed in Table 4:

OPTIMIZATION STRATEGY inTEGER COnTinUOUS OpTimizATiOn AppROACh
Th e particularity of our problem deals with integer and continuous variables simultaneously.Th e selection of SRM type for fi rst and second stages were considered as integer variables.
Meanwhile, the trajectory and design parameters of the third stage SRM were considered as continuous.
Several engineering applications of mixed integer continuous optimization approach were presented by Haupt et al. (2009), Faustino et al. (2006), as well as detailed explanation in Yeniay (2005) and Gantovnik et al. (2005).
Garfi eld and Allen (1995) used integer optimization applied to the configuration of LVs, Johnson (2002)  GA has been eff ectively applied to solve the problem of liquid propellant based LV (Riddle et al. 2007) solid propellant LVs (Bayley et al. 2008).Rafique et al. (2009) and Goldberg (1989) provides detailed and comprehensive implementation of GA in solving complex problems.

OpTimizATiOn mEThOD
The adopted and implemented GA optimization method is shown in Fig. 4, where a set of input design variables (SRM type, trajectory and third stage), as well as the lower and upper bounds, are passed to the main loop, where an initial population is randomly created.Furthermore, the selection, the crossover and the mutation operations are performed until the stopping criteria is achieved.The constraints were calculated and handled by external penalty function, as presented in Deb (2000) and detailed and explained in Coello (1999) and Kramer (2010).At each routine, the propulsion, mass, aerodynamics and trajectory analysis were performed.
The main characteristics of GA are presented in Table 5.

OpTimizATiOn FRAmEWORK
The optimization framework considered for this research is based on the multi-discipline feasible (MDF) design, which allows an easy and accurate result (Qazi and He L, 2006), as shown in Fig. 5.

OPTIMIZATION RESULT
The results show that the considered mixed integercontinuous GA optimization approach successfully reached the objective function.Th e optimized LV has a total mass of 23,530 kg and a 16.12 m of length.Table 6 shows the optimized value of variables and in Table 7 the main parameters of the LV third stage are listed.
The first and second stages SRM design type (SRM 12 and SRM 22), had been optimized and selected from Table 2.Both SRMs have the same diameter but diff erent length.As it is represented in Fig. 6, the shroud design is confi gured with the same diameter as the third stage, in order to reduce the aerodynamics forces and interferences.
The performance characteristics of the LV, shown in Fig. 7, demonstrates the capability of the three stage solid propellant LV to place a small payload into the LEO orbit maintaining its main parameters inside its limit values, furthermore, the overall design configuration facilitates its launch operations.

CONCLUSION
A small three-stage solid propellant LV was confi gured and optimized using a mixed integer-continuous GA optimization method.Th e fi rst and second stages SRM types were considered as integer variables, whereas the third stage SRM and trajectory as continuous.Th e main advantage of using GA relies on its independency of initial values to start the optimization, and the ability to handle integer variables.Th e propulsion, mass, aerodynamic and dynamic models were developed and integrated in a MDF framework.
LV is the LV gross mass, m 01 fi rst stage mass, m 02 second stage mass, m 03 third stage solid rocket mass, m IM instrument module mass, m PDM payload deployment module mass, m PAY payload mass, and m st the structural mass of the third stage SRM.

•Figure 1 .
Figure 1.Forces acting on a launch vehicle.

Figure 3 .
Figure 3. Pitch over ascent phase of launch vehicle.
conducted a screening process of booster for hypersonic vehicles, Calabro et al. (2002) presented the optimization of the propulsion for multistage LVs, and Bhatnagar et al. (2012) solved the mass distribution problem under restrictive condition.Hartfi eld et al. (2004) have shown the application of GA in fi nding the global optimum in ramjet propulsion.Bayley and Hartfi eld (2007) used GA for LV multidisciplinary design optimization with emphasis on minimum cost.

Figure 7 .
Figure 7. Performance characteristics of launch vehicle.

Table 3 .
, as well as Design variables.

Table 7 .
Parameters of launch vehicle third stage.

Table 6 .
Optimum values of variables.