Program to Optimize Simulated Trajectories II (POST2) Surrogate … · 2018-01-18 · Completion of the verification trials was then followed by surrogate model fitting. After fits

NASA/TM—2017–219842

Program to Optimize Simulated Trajectories II (POST2) Surrogate Models for Mars AscentVehicle (MAV) Performance AssessmentM.R. Zwack and P.D. DeesJacobs ESSSA, Marshall Space Flight Center, Huntsville, Alabama

H.D. Thomas, T.P. Polsgrove, and J.B. HoltMarshall Space Flight Center, Huntsville, Alabama

December 2017

National Aeronautics andSpace AdministrationIS02George C. Marshall Space Flight CenterHuntsville, Alabama 35812

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i

NASA/XX—2017–219842

Program to Optimize Simulated Trajectories II (POST2) Surrogate Models for Mars Ascent Vehicle (MAV) Performance AssessmentM.R. Zwack and P.D. DeesJacobs ESSSA, Marshall Space Flight Center, Huntsville, Alabama


December 2017

National Aeronautics andSpace Administration

Marshall Space Flight Center • Huntsville, Alabama 35812

ii

Available from:

NASA STI Information DeskMail Stop 148

NASA Langley Research CenterHampton, VA 23681–2199, USA

757–864–9658

This report is also available in electronic form at

iii

TABLE OF CONTENTS

1. INTRODUCTION ............................................................................................................. 1

2. APPROACH ....................................................................................................................... 2

3. GROUND RULES AND ASSUMPTIONS ....................................................................... 3

3.1 Design of Experiments ................................................................................................. 3 3.2 Program to Optimize Simulated Trajectories II Assumptions ....................................... 4

4. VERIFICATION ................................................................................................................ 5

5. SURROGATE MODELING ............................................................................................. 6

5.1 Model Fitting in JMP ................................................................................................... 6 5.2 Goodness-of-Fit Testing ............................................................................................... 7

6. RESULTS ........................................................................................................................... 9

REFERENCES ....................................................................................................................... 19

iv

LIST OF FIGURES

1. Percentage of error distribution for two-engine TSTO surrogate ............................... 8

2. Percentage of error distribution for three-engine TSTO surrogate ............................. 8

3. Liftoff latitude versus gross mass for two target orbits. Payload = 3 mt, Isp = 360 s, thrust = 100 kN, stage 1 PMF = 0.85, and stage 2 PMF = 0.75 ............... 10

4. Liftoff TW versus gross mass for two-engine MAV with stage 1 PMF = 0.8355, stage 2 PMF = 0.7639, crew cabin mass = 3 mt, latitude = 30°, and orbit = 1 Sol ....................................................................................................... 11

5. Liftoff TW versus gross mass for three-engine MAV with stage 1 PMF = 0.8355, stage 2 PMF = 0.7639, crew cabin mass = 3 mt, latitude 30°, and 1 Sol orbit ........................................................................................................... 11


7. Liftoff TW versus gross mass for three-engine MAV with stage 1 PMF = 0.8355, stage 2 PMF = 0.7639, crew cabin mass = 3 mt, latitude = 30°, and orbit = 5 Sol ....................................................................................................... 12

8. Liftoff TW versus gross mass for two-engine MAV with stage 1 PMF = 0.8507, stage 2 PMF =0.7898, crew cabin mass = 4 mt, latitude = 30°, and orbit = 1 Sol ....................................................................................................... 13




v

LIST OF FIGURES (Continued)

12. Payload versus gross mass for Isp = 335 s, latitude = 30°, and orbit = 1 Sol ............... 15






vi

LIST OF TABLES

1. DOE inputs and ranges for two-stage MAV .............................................................. 4

2. MultiPOST versus manual payload comparison ....................................................... 5

3. Excess delta-v values for target orbits ........................................................................ 9

4. Stage PMF settings for figures 12–17 ......................................................................... 15

vii

LIST OF ACRONYMS AND SYMBOLS

ACO Advanced Concepts Office

DOE design of experiments

MAV Mars Ascent Vehicle

PMF propellant mass fraction

POST Program to Optimize Simulated Trajectories II

RSE response to surface equation

Sol Martian solar day

TW thrust-to-weight

viii

NOMENCLATURE

Isp specific impulse

mb burnout mass

mp propellant mass

R response

x input parameter

β regression coefficient

1

TECHNICAL MEMORANDUM

PROGRAM TO OPTIMIZE SIMULATED TRAJECTORIES II (POST2) SURROGATE MODELS FOR MARS ASCENT VEHICLE (MAV) PERFORMANCE ASSESSMENT

1. INTRODUCTION

Trajectory optimization is a key component in assessing launch vehicle performance during conceptual design. For prephase A and phase A concept definition studies, the Integrated Space Transportation Team within Marshall Space Flight Center’s Advanced Concepts Office (ACO) implements Program to Optimize Simulated Trajectories II (POST2). POST2 is an industry-stan-dard tool that implements a direct solution method to approximate the control function for an optimal trajectory.1 Successfully employing POST2 for a trade study requires a trajectory expert in the loop to ensure that the control variables remain within the feasible space and produce accept-able trajectories.2 For this reason, trade studies can only be as broad as the trajectory expert’s time will allow. A very limited number of cases can be completed by hand in a given workday.

In an attempt to alleviate the execution time issues associated with POST2 and allow for much broader trade space exploration, ACO has developed a tool for automating ascent trajectory optimization. This tool captures heuristics developed over years of analyst experience and leverages the power of modern computing to speed up the evaluation of large sets of vehicle trajectories. The basic premise of the tool, known as multiple Programs to Optimize Simulated Trajectories, or mul-tiPOST, is to use multiprocessing to simultaneously evaluate random repetitions for a single vehicle case. The analysis-based heuristics are then used to help converge to acceptable trajectories or reject unrealistic ones. The approach implemented in multiPOST is detailed further in references 3 and 4.

Although primarily used by ACO for Earth-to-orbit vehicle studies, multiPOST is capable of executing POST2 for nearly any valid vehicle input deck. Using this flexibility, multiPOST was applied to carry out a Mars Ascent Vehicle (MAV) study. The MAV is a key component to success-ful human exploration of Mars and is being considered within NASA’s evolvable Mars campaign.5

The primary focus of this MAV study was to use multiPOST to investigate the effects of mass fraction, liftoff location, and propulsion on the payload mass delivered to a particular orbit by a two-stage ascent vehicle. The stages of the vehicle contain the propulsion related elements, such as engines and propellant tanks. The payload mass in this case was the crew cabin, which included the crew and provisions for the trip from the surface of Mars to rendezvous with the Mars-Earth transportation vehicle. The ultimate goal was to minimize the MAV gross mass for a given cabin mass. Any changes in the MAV mass will impact the sizing of other architecture ele-ments including Earth launch and in-space transportation as well as entry, descent, and landing.6

2

2. APPROACH

The primary purpose of the multiPOST tool is to enable the execution of much larger sets of vehicle cases to allow for broader trade space exploration. However, this exploration is not achieved solely with the increased case throughput. The multiPOST tool is applied to carry out a Design of Experiments (DOE), which is a set of cases that have been structured to capture a maximum amount of information about the design space with minimal computational effort. The results of the DOE are then used to fit a surrogate model, ultimately enabling parametric design space exploration.

The approach used for the MAV study includes both DOE and surrogate modeling. First, the primary design considerations for the vehicle were used to develop the variables and ranges for the multiPOST DOE. The final set of DOE variables were carefully selected in order to capture the desired vehicle trades and take into account any special considerations for surrogate modeling.

Next, the DOE sets were executed through multiPOST. Following successful completion of the DOE cases, a manual verification trial was performed. The trial involved randomly selecting cases from the DOE set and running them by hand. The results from the human analyst’s run and multiPOST were then compared to ensure that the automated runs were being executed properly. Completion of the verification trials was then followed by surrogate model fitting.

After fits to the multiPOST data were successfully created, the surrogate models were used as a stand-in for POST2 to carry out the desired MAV trades. Using the surrogate models in lieu of POST2 allowed for visualization of vehicle sensitivities to the input variables as well as rapid evaluation of vehicle performance. Although the models introduce some error into the output of the trade study, they were very effective at identifying areas of interest within the trade space for further refinement by human analysts.

The next section will cover all of the ground rules and assumptions associated with DOE setup and multiPOST execution. Section 3.1 gives the final DOE variables and ranges, while section 3.2 addresses the POST2 specific assumptions. The results of the verification trials are given in sec-tion 4. Section 5 gives the surrogate model fitting results, including the goodness-of-fit metrics for each fit. Finally, the MAV specific results are discussed in section 6.

3

3. GROUND RULES AND ASSUMPTIONS

3.1 Design of Experiments

Primary design considerations for the MAV include mission parameters such as surface location and destination orbit as well as vehicle parameters such as propulsion and vehicle masses. At this point in the MAV design, mass fractions are more appropriate due to the lack of detailed subsystem masses. Using a Propellant Mass Fraction (PMF) allows the analysis to be more flex-ible and to capture many unique vehicle configurations as long as they fit within the bounds of the study. The PMF for this study was defined for each stage as:

PMF = mp / (mp + mb) , (1)

where mp is stage usable propellant mass, and mb is stage burnout mass. The PMFs do not include any mass attributed to the crew cabin.

In addition to stage PMF, other vehicle parameters were added to the DOE to represent the MAV propulsion system. In order to capture many potential propulsion system types, continu-ous parameters were used for engine thrust and specific impulse (Isp). The ranges for thrust and Isp were developed to capture potential propellant combinations for the MAV (e.g. liquid oxygen/liquid methane, nitrogen tetroxide/monomethyl hydrazine). This allowed the analyst to dial values representing specific types of propulsion into a single surrogate model.

The variable for number of engines is discrete, which requires special treatment. Due to the discrete nature of this variable, the DOE sets were split to improve the ease of fitting surro-gate models. Discrete parameters typically cause difficulty in achieving acceptable fits, therefore a single surrogate is fit to each discrete dataset. It is important to note that the number of engines only applies to the first stage of the vehicle. It was assumed for this study that the second stage would always contain a single engine. This engine is assumed to be identical in thrust and Isp to the engines on the first stage.

Two mission parameters of interest were included in the study DOE. First, the destination orbits for the MAV were defined based upon current Mars orbit rendezvous options at 1 Sol and 5 Sol.6 These orbit options are discrete; however, they were represented using a continuous parameter for the required reserve delta-v. The ranges shown in table 1 for reserve delta-v were developed to capture orbit options under consideration for the MAV. The second mission parameter, latitude, is a continuous variable representing the liftoff location. Since no specific landing sites were identified for the study, the latitude range was set to capture a large number of landing sites that have been identified for consideration.7 The final set of DOE inputs and ranges for the two-stage MAV can be seen in table 1.

4

Table 1. DOE inputs and ranges for two-stage MAV.

Input Variable Min MaxNo. of engines (discrete) 2, 3Reserve delta-V (m/s) 1,200 1,800Liftoff latitude (°) 0 60Thrust per engine (kN) 68 200Engine Isp (s) 300 460Stage 1 PMF 0.75 0.95Stage 2 PMF 0.65 0.9Payload (crew cabin) mass (kg) 1,800 5,000

3.2 Program to Optimize Simulated Trajectories II Assumptions

The POST2 input deck used common assumptions for the Martian environment. The gravitational model for Mars comes from the 1997 Astronomical Almanac.8 The atmospheric data was taken from the Space and Planetary Environment Criteria Guidelines for Use in Space Vehicle Development.9 The reference area for MAV aerodynamics was set at 22.06 m2 with the speed of sound input as 487.89332 m/s.6

The POST2 input deck was set up to fly to an initial orbit of 100 by 250 km while reserving an excess delta-v required to transition to a higher target orbit. The excess delta-v required was set as the variable dvmarr in the deck.

The dependent variables in the POST2 input deck defined the target orbit using geocentric radius, flight path angle, and inertial velocity. Independent variables for optimization consisted of seven pitch rates, launch azimuth, initial weight of the payload, and throttle level for a throttling event during ascent. The throttle level was constrained to a minimum of 20%.

The first pitch event occurs at 5 s into the ascent, followed by a gravity turn at 45 s. Between the end of the gravity turn and insertion into the initial orbit, six additional pitch events occur. The gravity turn is ended by the first stage reaching zero propellant. At this point, the first stage is jet-tisoned, and the second stage begins thrusting at its optimized throttle level.

The final main-engine cutoff for the two-stage vehicle occurs when the MAV achieves its initial orbit of 100 by 250 km. Simultaneously, the MAV is constrained to have a remaining delta-v equal to the excess amount required to reach the final target orbit.

5

4. VERIFICATION

Using the variables and ranges in table 1, a DOE was executed through the multiPOST tool. Random cases were then selected from the DOE to test the multiPOST execution of the MAV POST2 deck. These cases were extracted from the results and individually checked by a POST2 expert. It is expected that the POST2 expert will beat multiPOST in most cases. If any large differ-ences (>3%) in output are noticed during these checks, the setup and execution of the automated deck is investigated. Table 2 shows payload comparisons for a representative set of cases from the MAV study.

Table 2. MultiPOST versus manual payload comparison.

ManualPayload

(kg)

AutomatedPayload

(kg)

Delta-V(Manual-Auto)

(%)4,818.55 4,818.12 0.0083,608.79 3,608.79 04,563.15 4,555.90 0.162,828.51 2,828.76 – 0.0093,977.55 3,978.55 – 0.0253,409.29 3,401.56 0.2264,447.39 4,428.31 0.4294,529.77 4,517.74 0.2664,145.21 4,119.40 0.6235,331.47 5,331.01 0.0084,041.84 4,041.66 0.0053,815.59 3,815.66 – 0.002

6

5. SURROGATE MODELING

5.1 Model Fitting in JMP

Following successful verification testing, surrogate models were fit for each of the discrete datasets. The statistical software JMP was used to complete the model fitting. This software pro-vides tools for fitting many types of models including Gaussian processes, Kriging, neural net-works, and response surface equations.10,11 Response Surface Equations (RSE) were selected as the desired model type for the MAV study due to their relative simplicity as compared to Gaussian process models or neural networks. Although the trajectory response space is typically multimodal and nonlinear, higher order RSEs have been used to successfully fit results from POST2 for other studies.3

RSEs are typically fit using the method of least squares to estimate the regression coeffi-cients.12 Equation 1 gives the generic form of a second order RSE, where R is the response, x terms are input parameters, and β terms are regression coefficients. Note that the model is linear in the β parameters and is therefore a linear regression model, regardless of the order of the input param-eters12:

R = β0 + β jx jj=1

k

∑ + βijxix jj=i+1

k

∑i=1

k−1

∑ + β jjx2jj=1

k

∑ . (2)

As noted above, higher order RSE models have been successfully used to fit POST2 output data. However, as the order of the model is increased, the total number of terms in the equation also increases. Since k + 1 cases are required to regress the β coefficients for a model of k terms, this also increases the total number of DOE cases that must be completed through multiPOST. In order to keep the number of required cases as well as the overall length of the equation to a reasonable level, stepwise regression was utilized in JMP.

Stepwise regression uses a statistical significance level to determine which terms in the regression model are most beneficial to the model’s ability to predict the data. During model fitting, terms are added or removed from the model based upon their calculated significance levels. The result is a model containing only terms that are statistically significant for predicting the response, which tends to be much more compact than the full model.

In addition to stepwise regression, k-folds cross validation was implemented when fitting the models in JMP. K-folds cross validation splits the full set of cases into k different sets or ‘folds.’ The model is fit to k − 1 sets of data, leaving the remaining set for model validation. After fitting a total of k models, the one with the best validation statistics is returned as the final model.10

7

5.2 Goodness-of-Fit Testing

When a surrogate model is returned, multiple goodness-of-fit tests are applied to accept or reject the fit. First, the coefficient of multiple determination, or R2, is used as an initial indicator of fit acceptability. The R2 value is a measure of the reduction in variability of the response obtained from the model.12 Based upon experience with fitting POST2 data, an R2 value greater than 0.99 warrants further goodness-of-fit testing. Any value below 0.99 will require refitting after investiga-tion of outliers or execution of additional DOE runs.

When fitting to POST2 data, the investigation of outliers tends to be the only action required to improve a fit. Although multiPOST utilizes heuristics to eliminate unreasonable trajec-tories, some of these trajectories can still pass along into the final data set. These cases are identi-fied using the residual-by-predicted plot in JMP. This plot shows the predicted response value for each point plotted against its residual as compared to the actual response value. The desired shape of this plot is a ‘shotgun spread’ of points with no clear trends—an example of which is shown in reference 3. Any points that are significantly outside the main point cloud are tagged as outliers for further investigation.

Using the detailed POST2 output for each of the outliers, an analyst can easily determine whether or not the points can be excluded from the fit. Typically, these cases illustrate odd behavior, such as extreme lofting (affectionately dubbed a ‘rollercoaster’ trajectory) or very excessive pitch profiles due to a high thrust-to-weight (TW). Any case deemed to be unrealistic or unflyable for a crewed MAV is then removed from the model fitting dataset.

The final goodness-of-fit test is used to calculate the error percentage of the surrogate model across the entire dataset. This percentage encompasses both the points used to fit the model itself and the points held back for validation purposes. After calculating the error, the characteristics of the percent error distribution are used as a final acceptance check of the model. Typically, the mean is desired to be as close to zero as possible and the standard deviation must be below one. Figures 1 and 2 provide the final error distributions for the MAV surrogate models. Note that the error per-centage was calculated based upon the difference between the surrogate model and the multiPOST output.

8

32.5

21.5

10.5

0– 0.5

–1–1.5

–2–2.5

–3–3.5

– 4

10099.597.5

9075502510

2.50.5

0

Maximum 3.076782.24699

1.36540.68433

Quartile 0.29347Median – 0.0153Quartile – 0.2758

– 0.6375–1.3016–2.6042

Minimum –3.8256

MeanStd DevStd Err MeanUpper 95% MeanLower 95% MeanN

0.00537230.63029910.01192860.0287621– 0.018017

2,792

Quantiles (%)

Summary Statistics

F1_1743Observance

Erro

r (%

)

Figure 1. Percentage of error distribution for two-engine TSTO surrogate.

33.5

2.52

1.51

0.50

– 0.5–1

–1.5–2

–2.5–3

–3.5–4

– 4.5–5

10099.597.5

9075502510

2.50.5

0

Maximum 3.468952.011681.037870.45785

Quartile 0.18612Median 0.01718

Quartile – 0.1725–0.496–1.989

–1.7967 Minimum – 4.7505

MeanStd DevStd Err MeanUpper 95% MeanLower 95% MeanN

0.00164690.49752310.0071484

0.015661– 0.012367

4,844

Quantiles (%)

Summary Statistics

F2_1743

Observance

Erro

r (%

)

Figure 2. Percentage of error distribution for three-engine TSTO surrogate.

It is important to note that the resulting surrogate models are only valid within the bounds shown in table 1. Any case within the bounds that is evaluated through the surrogate will lie within the error distributions in figures 1 and 2. If any of the input variables entered into the surrogates are outside the bounds, the response of the surrogate is not valid, and it is highly likely that the percent error will be well outside of the distributions shown in the figures.

9

6. RESULTS

After performing goodness-of-fit testing and accepting the MAV surrogates, the equations were used to analyze trades of interest. As discussed in section 3, these trades include propellant types, crew cabin (payload) masses, and stage PMFs. Propellant types were represented by varying the engine thrust and Isp. Multiple destination orbits were represented by fixing the excess delta-v input variable in the surrogate models. The excess delta-v values used for the specific orbits can be seen in table 3.

Table 3. Excess delta-v values for target orbits.

OrbitExcess Delta-V

(m/s)1 Sol 1,4345 Sol 1,580

First, the sensitivity to liftoff latitude was analyzed for each target orbit. Figure 3 shows an example of the latitude sensitivity for a representative two-stage MAV flying to two different target orbits. As seen in the figure, increasing latitude increases the gross liftoff mass by approx imately 2–3 metric tons over the entire range of latitude settings. This effect, however, is very small in comparison to the effects of thrust, Isp, and stage PMF, which can affect gross mass by tens of tons over their respective ranges. Therefore, the latitude was fixed at the median value of 30° for the rest of the results within this section.

10

31,000

30,500

30,000

29,500

29,000

28,500

28,000

27,500

27,000

26,500

26,000

25,500

25,0000 5 10 15 20 25 30 35 40 45 50 55 60

Gros

s Mas

s (kg

)

Latitude (°)

F3_1741

2 Engines, 1 Sol2 Engines, 5 Sol3 Engines, 1 Sol3 Engines, 5 Sol

Orbit and No. of Engines

Figure 3. Liftoff latitude versus gross mass for two target orbits. Payload = 3 mt, Isp =360 s, thrust = 100 kN, stage 1 PMF = 0.85, and stage 2 PMF = 0.75.

After fixing latitude, a first set of figures was developed to investigate the optimal thrust level for the two-stage MAV with a given crew cabin mass. As discussed previously, the Mars mis-sion architecture is very sensitive to the mass of the MAV. Optimizing the thrust level for a given payload will result in the smallest vehicle gross mass, thereby reducing the mass of the architecture.

To generate these figures, the thrust per engine was varied across the entire range noted in table 1 for both the two- and three- engine configurations. The PMFs for each stage were held constant using values representing a nominal MAV configuration. In figures 4–7, the first stage PMF was 0.8355, and the second stage PMF was 0.7639. The first stage PMF was 0.8507, and the second stage PMF was 0.7898 for figures 8–11. The variations in engine thrust were carried out at various Isp settings, which are colored in each figure. Figures 4–7 used a fixed crew cabin mass of three metric tons, while figures 8–11 used four metric tons. From these figures, an optimal TW can be identified for a given orbit and Isp.

11

45,000

42,500

40,000

37,500

35,000

32,500

30,000

27,500

25,000

22,500

20,000

17,500

15,0000.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Gros

s Mas

s (kg

)

Liftoff T/W

F4_1743

335360460

Isp

Figure 4. Liftoff TW versus gross mass for two-engine MAV with stage 1 PMF = 0.8355, stage 2 PMF = 0.7639, crew cabin mass = 3 mt, latitude = 30°, and orbit = 1 Sol.

45,000

42,500

40,000

37,500

35,000

32,500

30,000

27,500

25,000

22,500

20,000

17,500

15,00010.8 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Gros

s Mas

s (kg

)

Liftoff T/W

F5_1743

335360460

Isp

Figure 5. Liftoff TW versus gross mass for three-engine MAV with stage 1 PMF = 0.8355, stage 2 PMF = 0.7639, crew cabin mass = 3 mt, latitude 30°, and 1 Sol orbit.

12

45,00047,500

50,000

42,50040,00037,50035,00032,50030,00027,50025,00022,50020,00017,50015,000

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Gros

s Mas

s (kg

)

Liftoff T/W

F6_1743

335360460

Isp


45,000

47,50050,000

42,50040,00037,50035,00032,50030,00027,50025,00022,50020,00017,50015,000

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

Gros

s Mas

s (kg

)

Liftoff T/W

F7_1741

335360460

Isp

Figure 7. Liftoff TW versus gross mass for three-engine MAV with stage 1 PMF = 0.8355, stage 2 PMF = 0.7639, crew cabin mass = 3 mt, latitude = 30°, and orbit = 5 Sol.

13

45,00047,50050,00052,50055,00057,50060,000

42,50040,00037,50035,00032,50030,00027,50025,00022,50020,00017,50015,000

0.60.4 0.8 1 1.2 1.4 1.6 1.8

Gros

s Mas

s (kg

)

Liftoff T/WF8_1741

335360460

Isp

Figure 8. Liftoff TW versus gross mass for two-engine MAV with stage 1 PMF = 0.8507, stage 2 PMF =0.7898, crew cabin mass = 4 mt, latitude = 30°, and orbit = 1 Sol.

45,00047,50050,00052,50055,00057,50060,000

42,50040,00037,50035,00032,50030,00027,50025,00022,50020,00017,50015,000

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Gros

s Mas

s (kg

)

Liftoff T/W

F9_1743

335360460

Isp


14

45,00047,50050,00052,50055,00057,50060,000

42,50040,00037,50035,00032,50030,00027,50025,00022,50020,00017,50015,000

0.60.4 0.8 1 1.2 1.4 1.6 1.8

Gros

s Mas

s (kg

)

Liftoff T/W

F10_1743

335360460

Isp


45,00047,50050,00052,50055,00057,50060,000

42,50040,00037,50035,00032,50030,00027,50025,00022,50020,00017,50015,000

0.8 1 1.2 1.4 1.6 1.8 2 2.2

Gros

s Mas

s (kg

)

Liftoff TW

F11_1743

335360460

Isp


15

Using the optimal TW information, a second set of figures was developed to investigate the effects of various stage PMF settings on vehicle gross liftoff mass and payload delivered. This was done by varying the PMF settings of the vehicle stages while maintaining a near-optimal TW. Therefore, the data in the second set of figures consists of vehicles with optimal thrust-to-weights that produce a minimum gross liftoff mass for a given crew cabin mass.

The PMF settings represented worst, nominal, and best cases for the two-stage MAV and are listed in table 4. Figures 12–14 and figures 15–17 illustrate the payload delivered versus gross liftoff mass for various Isp and PMF assumptions for a 1 Sol and 5 Sol orbit, respectively.

Table 4. Stage PMF settings for figures 12–17.

Setting Name Stage 1 PMF Stage 2 PMFLow 0.75 0.65Nominal 0.85 0.75High 0.95 0.85

11,000

10,000

9,000

8,000

7,000

6,000

5,000

4,000

3,000

2,000

1,0002,000 2,500 3,000 3,500 4,000 4,500 5,000

Gros

s Mas

s (kg

)

Payload (kg)

F12_1743

PMF SettingLowNominalHigh

Figure 12. Payload versus gross mass for Isp = 335 s, latitude = 30°, and orbit = 1 Sol.

16

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19

REFERENCES

1. Steffens, M.J.; Edwards, S.J.; and Mavris, D.N.: “Capturing the Global Feasible Design Space for Launch Vehicle Ascent Trajectories,” Paper Presented at AIAA Atmospheric Flight Con-ference, AAIA SciTech Forum, Kissimmee, FL, January 5–9, 2015, doi:10.2514/6.2015-1910.

2. Steffens, M.J.; and Mavris, D.N.: Launch Vehicle Performance Analysis using Extreme Value Theory,” Paper Presented at AIAA SPACE Conference and Exposition, Pasadena, CA, August 31–September 2, 2015, doi:10.2514/6.2015-4679.

3. Dees, P.D.; Zwack, M.R.; Steffens, M.J.; Edwards, S.J.; Diaz, M.J.; and Holt, J.B.: “An Expert System-Driven Method for Parametric Trajectory Optimization During Conceptual Design,” Paper Presented at AIAA SPACE Conference and Exposition, Pasadena, CA, August 31–

September 2, 2015, doi:10.2514/6.2015-4486.

4. Dees, P.D; Zwack, M.R; Steffens, M.J.; and Edwards, S.J.: “Augmenting Conceptual Design Trajectory Tradespace Exploration with Graph Theory,” Paper Presented at AIAA SPACE Conference and Exposition, Pasadena, CA, September 13–16, 2016.

5. Percy, T.K.; McGuire, M.L.; and Polsgrove, T.T.: In-Space Transportation for NASA’s Evolv-able Mars Campaign,” Paper Presented at AIAA SPACE Conference and Exposition, Pasa-dena, CA, August 31–September 2, 2015.

6. Polsgrove, T.T.; Thomas, D.; Sutherlin, S.G.; Stephens, W.; and Rucker, M.: Mars Ascent Vehicle Design for Human Exploration,” Paper Presented at AIAA SPACE Conference and Exposition, Pasadena, CA, August 31–September 2, 2015.

7. Hoffman, S.J.; and Bussey, D.B.: “Human Mars Landing Site and Impacts on Mars Surface Operations,” Paper Presented at IEEE Aerospace Conference, Big Sky, MT, March 3–12, 2016.

8. U.S. Naval Observatory: The Astonomical Almanac, U.S. Government Printing Office, 1997.

9. Smith, R.E.; and West, G.S.: “Space and planetary environment criteria guidelines for use in space vehicle development, Volume 1: 1982 revision,” NASA/TR–82478, NASA Marshall Space Flight Center, Huntsville, AL, 244 pp., January 1983.

10. JMP: JMP 11 Fitting Linear Models, Version 11, SAS Institute Inc., Cary, NC, 478 pp.,2013.

11. JMP: JMP 11 Specialized Models, SAS Institute Inc., Cary, NC, 226 pp., 2013.

12. Myers, R.H.; Montgomery, D.C.; and Anderson-Cook, C.M.: Response Surface Methodology, John Wiley and Sons (3rd ed.), Hoboken, NJ, 856 pp., 2016.

20

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Standard Form 298 (Rev. 8-98)Prescribed by ANSI Std. Z39-18

Program to Optimize Simulated Trajectories II (POST2) Surrogate Models for Mars Ascent Vehicle (MAV) Performance Assessment

M.R. Zwack,* P.D. Dees,* H.D. Thomas, T.P. Polsgrove, and J.B. Holt

George C. Marshall Space Flight CenterHuntsville, AL 35812

National Aeronautics and Space AdministrationWashington, DC 20546–0001

Unclassified-UnlimitedSubject Category 15Availability: NASA STI Information Desk (757–864–9658)

Prepared by *Jacobs ESSSA Group and the Valves, Actuators, and Ducts Design and Development Branch, Propulsion Systems Office, Marshall Space Flight Center.

M–1445

Technical Memorandum

NASA/TM—2017–219842

Mars ascent vehicle, POST2, surrogate model, trajectory optimization, design of experiments

01–12–2017

UU 32

NASA

U U U

A Mars Ascent Vehicle (MAV) study was carried out using the Program to Optimize Simulated Trajec-tories (POST2) for performance assessment. An in-house tool called multiPOST was implemented to automate the execution of POST2. The results show the effects of vehicle mass fractions and propulsion parameters on vehicle gross liftoff mass and payload delivered. In addition to the results figures, surrogate models created from the POST2 output are included in a supplement file. These models encompass the trade space for the MAV study and can be used as a stand-in for POST2 when evaluating vehicle perfor-mance in terms of payload delivered.

STI Help Desk at email: [email protected]

STI Help Desk at: 757–864–9658

NASA/TM—2017–

Program to Optimize Simulated Trajectories II (POST2) Surrogate Models for Mars AscentVehicle (MAV) Performance AssessmentM.R. Zwack and P.D. DeesJacobs ESSSA, Marshall Space Flight Center, Huntsville, Alabama


October 2017

National Aeronautics andSpace AdministrationIS02George C. Marshall Space Flight CenterHuntsville, Alabama 35812

Blank Page

Instructions

POST2 Surrogate Models for Two-stage-to-orbit Mars Ascent Vehicle Analysis

Authors: Mathew R. Zwack, Patrick D. Dees, James B. Holt

This Excel document contains surrogate models for Mars Ascent Vehicle Performance as described in the NASA/TM "Program to Optimize Simulated Trajectories II (POST2) Surrogate Models for Mars Ascent Vehicle (MAV) Performance Assessment". The two surrogate models are contained in respective worksheets, "2Eng" and "3Eng" along with error bound calculations for each. In the surrogate tabs the user will find the vehicle inputs in columns A through G. Note, each input has a cell comment giving the allowable ranges for the respective input. Any inputs entered beyond these ranges will produce results that are NOT valid. The surrogate model equations are contained from columns K to QW (2Eng) or PA (3Eng) with the output in column J. The user is advised not to change any of these cells as any adjustments to the equation coefficients would render the equation invalid.

Basic Usage:Basic usage involves execution of single vehicle case. As can be seen in the worksheets, the first row has been pre-populated with a vehicle case. To analyze a different single vehicle the user simply changes one or more of the input values and the surrogate output will update in column J.

Multiple Cases:The surrogate equations were set up in row format to support easier execution of multiple vehicle cases at once. As shown in the 2Eng tab, three cases are being evaluated. To do this, the user can extend all of the surrogate model term columns (J to QW in this case) down multiple rows. By entering new values for the inputs multiple cases can be executed at once.

Note, for proper execution of the error bounds, the error bound calc sheet must be updated for additional cases. To update this sheet, go to the bottom of the covariance matrix (in 2EngErrorBoundCalcs this matrix is A2 to QM456). The three example cases can be seen starting in row 459. Steps are as follows:

1. Select all cells in the last row of the list (row 461 cols A through QM in this case) and extend the equations down additional row(s) for the desired number of cases.

2. Next, find the sumproduct column (QQ in this case) and extend the equation in the first cell (QQ2) to the right for the desired number of cases. Note the first array in the sumproduct equation should always read A2:QM2, while the second entry should correspond to the row number of the case in step 1 (i.e. 461 for the third example case)

3. Extend the equations in the new columns down to row 456

4. Extend the second sumproduct in row 457 to the right for each of the new columns and ensure the entries are correct. Note the first entry in the sumproduct should correspond to rows 2 through 456 in the respective column (i.e. for col QQ; QQ2:QQ456). The second entry should reference cells A through QM of the row number for the new case from step 1 (i.e. the third example case is A461:QM461). When entries are correct you must hit CRTL+SHIFT+ENTER to properly execute the equation.

5. Extend the multiplier cells (i.e. QQ458 and QQ459) to the right for the newly added columns to get the error bound in row 459

6. Ensure the newly calculated percentile number is being added/subracted correctly in the surrogate tab. For example, the third case (row 4 cols H,I) is pointing to the percentile number in QQ459. If a 4th case were added to the 2Eng tab, cols H and I would then point to QR459 after the steps above were completed.

2Eng

Payload (lbs)

Zwack, Mathew R. (MSFC-ED04)[ESSSA]: Zwack, Mathew R. (MSFC-ED04)[ESSSA]:Min: 4,000Max: 11,000Thrust per Engine (lbf)

Zwack, Mathew R. (MSFC-ED04)[ESSSA]: Zwack, Mathew R. (MSFC-ED04)[ESSSA]:Min: 15,000Max: 45,000Isp (s)

Zwack, Mathew R. (MSFC-ED04)[ESSSA]: Zwack, Mathew R. (MSFC-ED04)[ESSSA]:Min: 300Max: 460Stage1PMF

Zwack, Mathew R. (MSFC-ED04)[ESSSA]: Zwack, Mathew R. (MSFC-ED04)[ESSSA]:Min: 0.75Max: 0.95Stage2PMF

Zwack, Mathew R. (MSFC-ED04)[ESSSA]: Zwack, Mathew R. (MSFC-ED04)[ESSSA]:Min: 0.65Max: 0.9Latitude (deg)

Zwack, Mathew R. (MSFC-ED04)[ESSSA]: Zwack, Mathew R. (MSFC-ED04)[ESSSA]:Min: 0Max: 60dvmarr (ft/s)

Zwack, Mathew R. (MSFC-ED04)[ESSSA]: Zwack, Mathew R. (MSFC-ED04)[ESSSA]:Min: 4,000Max: 6,000Lower 95%Upper 95%Predicted Gross Mass (lbs)Surrogate Model Terms --->

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2EngErrorBoundCalcs

Covariance Matrix (from JMP)Sum Product (Case 1 on 2Eng tab)

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Program to Optimize Simulated Trajectories II (POST2) Surrogate … · 2018-01-18 · Completion of the verification trials was then followed by surrogate model fitting. After fits