Probabilistic Design of a Reusable Single-Stage-to-Orbit Launch ...

Probabilistic Design of a Reusable Single-Stage-to-Orbit Launch Vehicle Utilizing Parametric 3D CAD Model Patrick R. Chai

∗

ME6104 Fundamentals of Computer Aided Design

Georgia Institute of Technology, Atlanta, GA 30332

The design of a complex vehicle such as an orbiter is extremely time consuming due to the large number of design variables and constraints that must be considered. Traditionally, the design of such vehicle is done with point-to-point iteration, where each change in the design variable requires every subsystem to be recomputed to ”re-close” the vehicle. The vehicle geometry and packaging is often left as an afterthought in the design process, leaving it highly constrained to the design. To account for uncertainty in the design, probabilistic methods are employed to determine the sensitivity of the vehicle’s mass property. This gives insight into the variation of the mass of the vehicle, but it does not address the changes in vehicle geometry. A 3D parametric CAD model is developed concurrently with a parametric sizing model. The models are used in a Monte Carlo Simulation with uncertainty distribution in the sizing routine. The results of the simulation is fed into the CAD model and the resulting vehicle geometry variation is presented. ∗

Graduate Research Assistant, Aerospace Engineering, resident at the National Institute of Aerospace, 100 Exploration Way, Hampton, VA 23666

1 of 18

Acronyms Single Stage to Orbit MDO Multidisciplinary Design Optimization CAD Computer Aided Design CAE Computer Aided Engineering LOX Liquid Oxygen LH2 Liquid Hydrogen LVSSS Launch Vehicle & Spacecraft Sizing and Synthesis MER Mass Estimating Relationships POST Program to Optimize Simulated Trajectory GLOW Gross Lift-Off Weight SSTO

Nomenclature m0

mf ∆V Isp g0 T /W T /Weng µ σ

initial, gross liftoff mass final mass velocity change specific impulse standard gravity system thrust-to-weight propulsion system T /W uncertainty factor standard deviation

I.

kg kg m/s s 9.80665 m/s2

Introduction

For decades, the fully-reusable, single-stage-to-orbit (SSTO) rocket launch vehicle with aircraft-like operations has been a prevailing vision for affordable, safe, and routine space access.1 However, at current technology levels, the Tsiolkovsky rocket equation limits SSTO vehicle performance to impractical mass fractions. All launch vehicles to date have employed multiple stages in order to overcome this limitation, and the only operational vehicle with reusable rocket elements has been the Space Shuttle. The design of a launch vehicle is extremely complex and time consuming due to the large number of design variables and constraints that must be considered. The design methodology

2 of 18

for such a vehicle has evolved with the advancement in technology and improved design tools. This evolution began in the early 1960s when experimentation is the primary tool for analysis due to the lack of knowledge in the theory. The age of experimentation was followed by physics-based modeling, simulation-based design and multidisciplinary design optimization (MDO) with the development of faster computers.2 MDO is widely used currently for designs of most launch systems which involve multiple domains such as structure, aerodynamics, and propulsions. To achieve an optimal design, the vehicle must be treated as a complete system instead of separate independent development of individual subsystems.3 To enable the framework of a multidisciplinary analysis tool, there is a need of an initial geometry model; however, the geometry model itself is depended on the result of the analysis tool.3 Most design overcome this paradox by defining a reference geometry, typically based on historical analysis or design, and modifying the geometry as necessary to fit the current design iteration. This approach creates “static” vehicle geometry that is very difficult to update and does not give real time feedbacks to decision makers. To improve the current MDO framework, the use of Computer Aided Design (CAD) /Computer Aided Engineering (CAE) can be the next evolutionary step in design methodology.4 CAD/CAE has been used in the design of automobile and aircraft components extensively,5678 but application to designs of launch vehicle has been less successful due to the extreme complexity of the problem. The complexity is further hampered by the uncertainty in the modeling of the subsystems. Typically, to account for the uncertainty in the model, probabilistic design methods are employed to determine the sensitivity of the vehicle’s mass property. This gives insight into the variation of the mass of the vehicle, but it does not address changes in vehicle geometry. To be able to visualize how changes in design variables affect the vehicle geometry will help designers anticipate potential problem areas that require detailed analysis. In this paper, a parametric 3D CAD model for a SSTO orbiter is developed. The parametric model is coupled with a MDO sizing and synthesis tool to evaluate the effects of uncertainty in modeling parameters on vehicle mass and geometry. The development of the sizing and synthesis tool and the parametric CAD model are documented. Uncertainty distri3 of 18

bution are applied to the synthesis tool and Monte Carlo probabilistic design are performed on the vehicle and the resulting vehicle geometry are presented and discussed.

II.

Modeling & Simulation

The vertical take-off/horizontal landing, LOX-LH2 , SSTO vehicle described in Ref. 9 and depicted in Figure 1 was selected as a baseline vehicle for this study. The baseline vehicle was originally designed in 1977 as part of an extensive study for NASA to determine the technology requirements for a post-Shuttle, fully reusable, SSTO launch vehicle, to enter service in 1995, and is taken to be representative of such a vehicle that could be built with evolutionary advances in current technology.

Figure 1. General arrangement of the baseline SSTO Vehicle9

Launch Vehicle & Spacecraft Sizing and Synthesis Tool The Launch Vehicle & Spacecraft Sizing and Synthesis (LVSSS)10 tool is created to calculate and size the mass and geometry of the vehicle. LVSSS tool is base on the concept presented in Ref. 10. The tool contains mass estimating relationships (MER) for the major subsystems of the vehicle. These MERs are generated from regression of historical data on the subsystems. All the MERs are created in a way that relates the subsystem sizing to fundamental engineering theories. For example, the aerodynamic surface is sized using historical data from aircrafts and spacecrafts, with the wing mass as a function of the vehicle’s mass, the wing’s span, surface area, root thickness, and the ultimate load factor. This relationship between the wing’s mass and the variables listed can be traced back to simple beam theory 4 of 18

and the Euler-Bernoulli beam equation. With the subsystem masses defined, the vehicle is integrated with the use of several similarity parameters. Rocket vehicle performance is governed primary by the rocket equation, m0 ∆V = exp( ) mf g0 Isp

(1)

where m0 and mf are the initial and final masses of the rocket stage, ∆V is the velocity change imparted to the rocket stage during the burn, Isp is the specific impulse of the rocket engine, and g0 is the standard gravitational acceleration. The initial mass, also referred to for launch vehicles as the gross liftoff mass, is the sum of the ascent propellant mass, the payload mass, and all other vehicle mass (called the inert mass minert ). The inert mass is further divided into the propulsion system mass, which is governed by the engine thrust-to-weight parameter T /Weng and the required system thrust, and the mass of the remaining structures, subsystems, on-orbit propellant, crew, equipment, etc. ∆V for the vehicle is determined by the reference mission, which in this case is a eastward launch from Kennedy Space Center into a 28.5◦ inclination, 93× 186 km orbit, with payload of 29,500 kg and an overall system T /W ratio of 1.3 at liftoff. The ideal ∆V , accounting for gravity, drag, and thrust losses is computed using regression analysis from data gathered from the Program to Optimize Simulated Trajectory (POST) developed by NASA and Martin Marietta Company.11 Specifying Isp and ∆V , the vehicle’s mass ratio is determined from equation 1 and an iteration scheme is used to converge the propellant required and the propellant available. The result from the iteration is a detail mass breakdown of the vehicle. A comparison between the baseline vehicle and the model prediction from LVSSS is show in Table 1.

Parametric 3D CAD Model The modeling of the SSTO vehicle began with making several simplifications in the model. The original model, as depicted in Ref. 9 and shown in Figure 1, includes the major and

5 of 18

Table 1. Abbreviated mass break down of Baseline Vehicle9 and LVSSS Model

System Aerodynamic Surfaces Body Structure Thermal Protection Main Propulsion Other inert mass (avionics, power, crew, reserves, losses, margins, etc) Subtotal - inert mass Ascent Fuel Ascent Oxidizer Payload Subtotal - GLOW

Baseline, kg 28,767 52,873 39,432 44,371 68,729

LVSSS, kg 30,695 56,809 41,042 40,217 67,957

% Diff 6.70% 7.44% 4.08% -9.36% -1.12%

234,172 207,625 1,453,373 29,484 1,924,654

236,720 220,810 1,545,676 29,484 2,032,690

7.26% 6.35% 6.35% 0.00% 5.61%

minor subsystems of the vehicle and packaged into the baseline design. To be able to create a parametric model that is scalable, the model needs to be simplified. The parametric CAD model created for this study is broken in to three major components: Body, Wing, and Tail. The three components of the vehicle is created using geometry information calculated in LVSSS for each design condition. Thirteen geometric variables are used in the definition of this simplified vehicle model, these variables are listed in Table 2. Table 2. Geometric Variable

Wing Span Root Thickness Root Chord Sweep Angle Dihedral Angle

Body Max Body Width LOX Tank Radius Payload Radius Max Body Height Body Length

Tail Tail Root Chord Tail Root Thickness tail Height

In the LVSSS tool, the geometric variables are calculated using either volume requirements, or by photographically scaling of the vehicle. The wing variables are photographically scaled to estimate the necessary aerodynamic surface area to produce required lift at landing. The tail is also photographically scaled in similar fashion. The body variables are scaled to match the propellant volume requirement of the vehicle since the majority of the enclosed volume is used for propellant storage. Typically, the payload bay radius is held

6 of 18

constant, however, considering aerodynamic properties of the vehicle, this variable is scaled with the rest of the body variable to give a more appealing design. The 3D model of the vehicle is created using SolidEdge. The wing, body, and tail were created as separate parts, and an assembly file is used to assemble this parts into the vehicle. The geometry of the orbiter are extracted from the dimensioned 2D drawings of Ref. 9 with approximations and simplifications made when necessary.

Wing

(a) Cross Section of Wing at Root

(b) Cross Section of Wing at Tip Figure 2. Parametric Geometry of the Orbiter Wing

The wing of the orbiter is created using an lofted extrusion between two sketched surface, the airfoil shape at the root and the airfoil shape at the tip. The sketch of the airfoil shape at the root is shown in Figure 2(a) and the sketch of the shape at the tip is shown in Figure 2(b). The shape of the airfoil is comprised of two bezier curve12 with four control vertices each. The two bezier curves are joined together at the maximum thickness with G1

7 of 18

continuity. The two curves are connected on the bottom of the airfoil with a straight line with G1 continuity at the leading edge and G0 continuity at the tailing edge. The geometric variables directly control the location of the control vertices and the length of the straight line. The location of the tip sketch is displaced by the span length in the span-wise direction, by a function of the dihedral angle and span in the vertical direction, and by a function of the swept angle and the span in the body length-wise direction. The solid rendering of the wing part is shown in Figure 3, where the blue outlines are the cross sections used for the lofted extrusion.

Figure 3. Solid model of the Orbiter Wing

Body The body of the orbiter is comprised of a series of lofted extrusion between several different cross-section sketches defined that are driven by the variables defined in Table 2. Three primary cross sections are shown in Figures 4(a), 4(b), and 4(c), note the sketches are not on the same scale. The rear cross section of the orbiter consists of a lower bay that is defined by the maximum body width and the LOX tank radius as defined by volume requirement and an upper bay that is defined by the payload bay radius. The rear end of the orbiter has this constant cross section for approximately a quarter of the total body length, as defined by the LOX tank length requirement. The shape of the body then contracts to the shape shown in Figure 4(b), where the measurements are taken from the drawings in Ref. 9. The sketch o the shape is taken to have as many G1 continuity as possible to ensure a smooth geometry

8 of 18

(a) Aft Orbiter Cross Section

(b) Orbiter Cross Section at 1/3 Body Length

(c) Cross Section of Nose Cone

(d) Solid Model of the Orbiter Body

Figure 4. Parametric Geometry of the Orbiter Body

and to avoid potential problems with scaling. Finally, the body of the orbiter contracts to a single circular cross section as defined by the nose cone diameter. The variables in the front of the orbiter are driven by fuel volume requirement as computed by LVSSS. The resulting solid model from the series of extrusion is shown in Figure 4(d). The extrusion on the top rear of the body is the connecting surface needed for the tail. This extrusion is scaled with the base of the tail.

Tail The sizing of the tail is very similar to the sizing of the primary wing. It is comprised of a lofted extrusion between two cross sections shown in Figures 5(a) and 5(b). These two cross sections define only half of the vertical tail, the part is mirrored on the axis that runs the length of the chord. The variables used to sized the tail were photographically scaled from the baseline vehicle to estimate the necessary wing area for stability and aerodynamic control. The shape of the tail is taken from dimensioned drawings in Ref. 9. The variables define the

9 of 18

(a) Tail Cross Section at Root

(b) Tail Cross Section at Tip

(c) Cross Section of Cutout

(d) Solid Model of Orbiter Tail

Figure 5. Parametric Geometry of the Orbiter Tail

10 of 18

location of control verticies of two separate bezier curve with G1 continuity between the two curve. Like the wing, the two bezier curve are connected by a straight line, this time with G0 continuity at both the leading edge and the trailing edge. To ensure continuity at the leading edge with the mirror function, the leading edge of the bezier curve is perpendicular to the straight line connecting the curves. The real tail of the orbiter as shown in Ref. 9 comprised of the main tail stricture with the rudder attached to the back. To simplify this, the tail is modeled as a single piece, with a cutout at the bottom to conform with the shape in the reference. The cut out, shown in Figure 5(c), is defined by a percentage of the tail height and root chord length as estimated from the drawings of the baseline vehicle. The resulting solid model of the tail is shown in Figure 5(d). The baseline assembled vehicle is shown in Figure 6.

Figure 6. Solid Mode of the Assembled Vehicle

Probabilistic Design To evaluate the effect of model and performance uncertainty on the vehicle’s overall geometry, a Monte Carlo analysis approach is taken.13 An uncertainty factor µ was multiplied to major subsystems to introduce variability to the LVSSS sizing result. The system recomputes and reiterates each case with the applied uncertainty factor to give a closed vehicle. For this

11 of 18

study, ten µ factors were used as design variables to vary the vehicle’s sizing characteristics. These variables are applied to the primary subsystems in the vehicle and are tabulated in Table 3. A 5,000 case Monte Carlo simulation was conducted using Phoenix Integration’s Modelcenter software. For each case, each µ is sampled independently from a triangular distribution, shown in Figure 7 (a = 0.8, b = 1.2, c = 1.0).14 The chosen distribution gives a ±20% spread to the nominal value of the subsystem. It should be noted that the LVSS model is used here to make predictions of vehicles with characteristics deviating significantly from the baseline vehicle and from any of the source vehicles comprising the data points upon with the subsystem MERs were generated. However, we are primarily interested in discovering trend information, and the absolute magnitude of the results is less important. Table 3. List of Uncertainty Factors used in Monte Carlo Simulation

µpropulsion µaero µbody µprotection µpower µconversion µguidance µcomm µinstrument µenvironment

Applied Applied Applied Applied Applied Applied Applied Applied Applied Applied

to to to to to to to to to to

all propulsion elements aerodynamic surfaces primary structures (Propellant tanks, thrust structure, landing gear, etc.) the thermal protection system sizing of the fuel cell units power conversion units guidance and navigation system communication system avionics and instruments environmental control system

Figure 7. Triangular Distribution

The results from the Monte Carlo Simulation are 5,000 data points containing each of the µ values and the resulting mass and geometry information from LVSSS. To post-process 12 of 18

these data, two programs are employed. SAS Institute Inc.’s JMP software is used to extract statistical information from the resulting data. The statistical information is extracted and inputted into the original parametric models to give different configurations for the vehicle at different confidence level. Screen shots of the parametric model are stitched together in Adobe’s Photoshop program and shown in the results section.

III.

Results and Discussion

Table 4. Statistical Information on Geometric Variables (units are in meters)

Wing Span Wing Thickness at Root Wing Chord Length at Root Body Width Body Length Body Height LOX Tank Radius Payload Radius Tail Thickness Root Tail Chord Length at Root Tail Height

mean 20.06 1.833 22.96 20.80 12.90 58.12 4.597 3.037 3.671 15.69 18.12

σ 2.744 0.2506 3.141 2.845 1.764 7.949 0.6287 0.4154 0.5021 2.146 2.479

max 31.01 2.833 35.50 32.15 19.34 89.83 7.015 4.695 5.674 24.26 28.01

90th 24.09 2.201 27.57 24.98 15.49 69.78 5.520 3.647 4.408 18.84 21.76

75th 21.73 1.985 24.88 22.54 13.98 62.96 4.980 3.291 3.978 17.00 19.64

25th 17.91 1.636 20.50 18.57 11.52 51.88 4.104 2.712 3.277 14.01 16.18

min 15.67 1.432 17.94 16.25 10.08 45.41 3.592 2.373 2.868 12.26 14.16

Vehicle Inert Mass (kg)

238,140

87,804

745,277

360,749

278,485

174,485

117,298

Table 4 tabulates the results of the Monte Carlo simulation. The columns of the table lists the mean, standard deviation, maximum, 90th percentile, 75th percentile, 25th percentile, and the minimum of each of the geometric variables as result from the 5,000 case simulation. As the table shows, the ±20% variation in the µ factor resulted in significant variation in both the geometry and the mass of the vehicle. This indicates that vehicle is highly sensitive to the model uncertainty. The large gap between the 90th percentile and the maximum (or 100th percentile) of the data also suggest that the data is not linear. Small uncertainty in the model is amplified in the resluting vehicle. The exponential nature of the rocket equation, combined with the uncertainty in the subsystem mass, is the source of great variability in the mass of SSTO vehicles at current technology levels. These effects are responsible for the

13 of 18

difficulties historically encountered in SSTO vehicle design and development. Figure 8 shows wire frame of the orbiter geometry using geometric variables at statistical percentiles. Each wire frame indicates the vehicle geometric requirement to account for that percentage of the design points. For instance, the vehicle outlined in blue will include seventyfive percent of the design space. As the figure shows, there is a significant gap between the 75th percentile and the 95th percentile. This gap is the result of the exponential nature of the rocket equation as discussed previously. It is interesting to note that the exponential nature of the rocket equation doesn’t become apparent until the higher percentile. The growth of the vehicle is fairly linear in the lower percentile in comparison. Another way of viewing the statistics is in terms of percent confidence. The model in this study has a ±20% uncertainty in the MERs of the major components. The Monte Carlo Simualtion results show the mass and geometric requirement to achieve certain confidence level in the vehicle design. In other word, the 75th percentile mass and geometry translates to a seventy-five percent confidence level in the design. This gives insight into how the uncertainty in the model impact the overall system level metrics. To achieve high level of confidence in the design, the designer must either reduce the uncertainty in the model to more accurately predict the system level performance, or the designer will have to include significant margin in the design to allow for vehicle growth. It is standard practice in system studies to include mass margin in designs. Depending on the maturity of the technology used in the design, the mass margin can range anywhere between 10 to 35 percent of the total dry mass. The use of mass margin allows designers to have more flexibility with the design but it does not address the fundamental issue of the model uncertainty. Also, mass margin does not address any geometric concerns of the vehicle. As the result of the Monte Carlo Simulation show, the ±20% uncertainty in the model can have signicant impact on the overall system. To achieve 90 percent confidence in the design, the mass of the vehicle grows 51% compare to the baseline, and the geomtry of the vehicle grows on average between 20 to 25 percent of the baseline. The typical mass margin applied to system design pales in comparison the actual vehicle growth in this study. The 14 of 18

assumption of 20 percent uncertaity is also very optmitistic, as most advanced technolgies will likely have even higher uncertaity level. All of these results points towrads a need of better understanding in the theory behind the advanced technologies and better models to predict the performance of these systems so the uncertainty can be reduced.

(a) Top

(b) 3D

(c) Front

(d) Side

Figure 8. Orbiter Wire Frame Geometry at 95th (Red), 75th (Blue), 50th(Black), and 25th(Green) Percentile

IV.

Intellectual Questions

In the aerospace industry, there is a movement to push CAD/CAE to the frontier of MDO framework as described by Staubach.4 The difficulty in a MDO framework is for all the separate disciples to work in unison with overall system level goals as the primary focus. This is very challenging, because system level goals typically does not coincide with subsystem level objectives. The CAD/CAE framework is useful especially in conceptual design because it brings system level integration into the analysis much earlier in the design

15 of 18

process. Instead of each individual discipline doing independent analysis of their field, the CAD/CAE framework will bring the analysis together. With a parametric model, it will also give the project managers insight into the sensitivity of the design and can allocate appropriate resources to areas of concern. The most beneficial aspect of utilizing CAD/CAM in the MDO framework is that by creating a detailed model of what ever system the MDO is studying, it gives the subsystem level experts an idea of how their system fits into the overall scheme. It also helps the project manager to predict potential integration issues so that they can be examined in detail. There are several discipline that require geometry information to perform their analysis (structure, aerodynamics, ground operations, etc), a unified CAD environment will give all the disciplines a single geometry to work from so that the analysis will be uniform and the uncertainty can be reduced.

V.

Conclusions

A parametric CAD and sizing model of a SSTO Orbiter was created in order to investigate the effects of uncertainty in the sizing model on overall system metrics. The exponential nature of the rocket equation amplifies the uncertainty in the sizing model and results in significant variation in the vehicle geometry and mass. Traditional mass margins used in system level design does not account for the impact of uncertainty on the vehicle geometry and is typically not enough to give a high level of confidence in the design. Application of mass margin does not address the fundamental problem of uncertainty in the modeling of the subsystems. In order to have high confidence in the design, large mass margin is required and usually makes the design infeasible. To improve the confidence of the design without large mass margin, the uncertainty in the modeling must be improve. Utilizing CAD/CAM within the MDO framework will allow project managers to identify the areas of highest sensitivity and put additional resources to improves the area of greatest uncertainty. By tackling the root of the problem, the need for mass margin can be reduced while the design become more accurate. Geometric and mass property of the vehicle has to be integrated into the MDO

16 of 18

framework concurrently.

VI.

Future Work

The CAD model developed is grossly simplified to fit into the framework of this study. In order to understand higher level interaction between the subsystem and the overall geometry of the vehicle, a higher fidelity level model of the orbiter is needed. The new model will need to have actual propellant tanks which are driven by the volume requirements of the vehicle, propulsion elements that are driven by the trajectory, and aerodynamic surfaces that are driven by actual aerodynamic data instead of photographic scaling. Integration of trajectory model into the LVSSS model is also required to improve the fidelity of the LVSSS model, and more detail break down of the vehicle is needed to determine the interaction of sub-subsystem level components. To improve the uncertainty analysis, the uncertainty distribution of the MERs needs to be improved. A combination of theory, research of past studies, and expert elicitation should be performed to determine the uncertainty bounds for each MERs. Monte Carlo Simulation should be performed with the new uncertainty bounds and new LVSSS model to produce more detailed geometric variations of the vehicle.

Acknowledgment This work is done in conjunction with Georgia Institute of Technology Mechanical Engineering course in Fundamental of Computer Aided Design instructed by Dr. David Rosen. The author would like to thank Dr. Rosen for his assistance in the completion of this paper.

References 1

Butrica, A. J., Single Stage to Orbit: Politics, Space Technology, and the Quest for Reusable Rocketry, The Johns Hopkins University Press, Baltimore, MD, 2003. 2 Tam, W. F., “Improvement Opportunities for Aerospace Design Process,” Space 2004 Conference and Exhibit, San Diego, California, AIAA 2004-6126, Sept. 2004. 3 Tarkian, M., Olvander, J., and Berry, P., “Exploring Parametric CAD-models in Aircraft Conceptual Design,” AIAA 2008-2162, April 2008. 4 Staubach, J. B., “Multidisciplinary Design Optimization, MDO, the Next Frontier of CAD/CAE in the Design of Aircraft Propulsion Systems,” International Air and Space Symposium and Exposition, Dayton, 17 of 18

Ohio, AIAA 2003-2803, July 2003. 5 Botkin, M. E., “Strutural Optimization of Automotive Body Components Based Upon Parametric Solid Modeling,” 8th AIAA/USAF/NASSA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, California, AIAA 2000-4707, Sept. 2000. 6 Mastin, C. W., Smith, R. E., Sadrehaghighi, I., and Wiese, M. R., “Geometric Model for a Parametric Study of the Blended-Wing-Body Airplane,” AIAA 1996-2416-cp, 1996. 7 Blair, M., “A demonstration of CAD/CAM/CAE in a fully associative aerospace design environment,” 37th AIAA/ASME/ASCE/AHS/ASC Structural Dynamics, and Materials Conference and Exhibit, Salt Lake City, Utah, AIAA 1996-1630, April 1996. 8 Wehrman, M. D., “Productivity Improvements Through the Use of CAD/CAM,” Journal of Aircraft, Vol. 22, No. 11, 1985, pp. 1013–1017, dot:10.2514/3.45240. 9 Haefeli, R. C., Littler, E. G., Hurley, J. B., and Winter, M. G., “Technology Requirements for Advanced Earth-Orbital Transportation Systems,” Final Report, NASA CR-2866, Oct. 1977. 10 Wilhite, A. W., Bush, L. B., Cruz, C. I., Lepsch, R. A., Morris, W. D., Stanley, D. O., and Wurster, K. E., “Advanced Technologies for Rocket Single-Stage-to-Orbit Vehicles,” Journal of Spacecraft and Rockets, Vol. 28, No. 6, 1991, pp. 646–651, dot:10.2514/3.26294. 11 Brauer, G. L., Cornick, D. E., Habeger, A. R., Petersen, F. M., and Stevenson, R., “Program to Optimize Simulated Trajectories (POST). Volume 1: Formulation Manual,” Final Report, NASA CR-132689, April 1975. 12 Zeid, I., Mastering CAD/CAM , McGraw-Hill Science/Engineering/Math, Columbus, OH, 2004. 13 McCromick, D. J. and Olds, J. R., “A Distributed Framework for Probabalistic Analysis,” 9th AIAA/SSMO Symposium on Multidisciplinary Analysis and Optimization, Atlanta, Georgia, AIAA 2002-5587, Sept. 2002. 14 Hayter, A. J., Probability and Statistics for Engineers and Scientists, Duxbury Press, Boston, MA, 2006.

18 of 18