November 2001

NASA/TM-2001-211246

An Approach to Simulation of Extreme

Conditions for a Planetary Lander

Eric M. Queen, Scott A. Striepe, and Richard W. Powell

Langley Research Center, Hampton, Virginia
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November 2001

NASA/TM-2001-211246

An Approach to Simulation of Extreme

Conditions for a Planetary Lander

Eric M. Queen, Scott A. Striepe, and Richard W. Powell

Langley Research Center, Hampton, Virginia
Available from:

NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS)

7121 Standard Drive 5285 Port Royal Road

Hanover, MD 21076-1320 Springfield, VA 22161-2171

(301) 621-0390 (703) 605-6000
An approach to simulation of extreme

conditions for a Planetary Lander

Eric M. Queen

Scott A. Striepe

Richard W. Powell

1 Abstract

A \Monte Carlo-like" design analysis tool is de- veloped and applied to an aeromaneuvering Mars

entry vehicle. This tool provides realistic but challenging design cases using many fewer cases than

a full Monte Carlo analysis. The problem of random input variables that are provided a priori (as

opposed to being drawn from a given distribution)

is addressed and a solution is found that shows

prospects for future improvement.

2 Introduction

The engineering design of a 
ight vehicle is often based on a \worst case" combination of external

conditions and manufacturing defects which the vehicle must be able to overcome to achieve its mission.

A better name for this case might be a \design case",

because a worse case can always be imagined, though

perhaps not a realistic one. Thus, a balance must be

maintained between designing to a case that is too

severe, leading to overdesign (higher weight, higher

cost) and a design case that is not severe enough and

sub jecting the vehicle to a failure that could have

been prevented. One approach that addresses this

issue is the use of Monte Carlo analysis as a design

tool. In this approach, one determines a realistic

range of values for each variable that will aect the

nal design (e.g. atmospheric dispersions, materials defects, measurement errors, etc.) and simulates

the system with many dierent combinations of input variables (typically thousands.) This design

approach has been used for the entry guidance system design of the Mars 2007 \Smart" Lander.

A disadvantage of Monte Carlo design is that

the extreme cases are the cases that drive the design.

Typically, a system performs very well for conditions

that are near the nominal, but not as well for conditions that are far from the nominal. For a large

number of inputs, many cases are near the nominal,

because relatively few cases fall far from the mean

and of those, fewer still occur in variables that are

strong contributors to the nal output. Thus, in

order to get a sucient number of extreme cases,

many thousands of Monte Carlo cases often need to

be run. The goal of the current work is to demonstrate a technique for simulating extreme cases that

are consistent with the Monte Carlo analysis, but

requires fewer runs.

The Monte Carlo simulation used in this study

is similar to that described in [Striepe et al], except

that the vehicle model has been changed to re
ect

the new mission. Notice that not all of the inputs

have the same type of distribution. Some are Gaussian, some are Uniform and some are provided as-is

and are not necessarily represented by any standard

distribution. Notably the initial position and velocity are results of another random process based on

the interplanetary tra jectory and are provided by

the NASA Jet Propulsion Laboratory (JPL).

3 Background

The vehicle used in this study is the proposed

2007 \Smart" Mars Lander. Of primary interest

will be the portion of the mission that involves

entry from interplanetary approach orbit until the

parachute is deployed near Mach 2. This vehicle is a

blunt body with a heatshield forebody very similar

to the Viking and Pathnder forebodies.

The purpose of the onboard guidance system is

to direct the vehicle safely to a predetermined target point. The ability of the guidance system to do

this is strongly aected by the interplanetary navigation; that is, the accuracy with which the spacecraft

achieves its target atmospheric entry condition. Two

types of navigation were considered. In the rst,

the standard radiometric measurements by the Deep

Space Network were assumed. In the second, additional optical measurements of Phobos and Deimos

1
by the onboard star tracker are incorporated, providing increased accuracy. The rst data type is

referred to as \radiometric" and the second as \optical". For both types of navigation, sets of 100,000

delivery positions and velocities were provided by

JPL. Additionally, an estimate of the vehicles position and velocity as determined for use by onboard

navigation systems was provided by JPL.

Since Pathnder,1the authors have assumed

that 2,000 cases were adequate to characterize

Monte Carlo results. But over the last several years,

the simulations have become more complex and have

more input variables. Also, as condence in the

methods increased, higher order statistics were utilized from the simulations. The concern is that

2,000 cases may be insucient for statistical analyses. Therefore, a Monte Carlo with 100,000 cases was run to determine the minimum number of cases

required for reliable statistics.

Figure 1 shows the mean true range to target

and mean navigated range to target for the radio0 1 2 3 4 5 6 7 8 9 10

x 104

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Number of Monte Carlo Cases

Mean Range

Mean True Range

Mean Nav Range 

Fig. 1. Radiometric Mean Range to target at

Parachute Deploy

metric only data as a function of number of cases

run. Both means are very well behaved and reach a

steady state after about 15,000 cases.

Figure 2 shows the standard deviations of range

to target for the radiometric only data, true and navigated. These statistics are much more volatile, with

10% changes after 50,000 cases. To some extent this

result is to be expected, since the standard deviation

is a higher order statistic than the mean.

Figures 3 and 4 show the same statistics as gures 1 and 2 but for the optical data set. Here

the standard deviation is slightly slower to converge

than the mean, but both are settled by 12,000 cases.

It is not clear why the standard deviation of the op0 1 2 3 4 5 6 7 8 9 10

x 104

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Number of Monte Carlo Cases

Standard Deviation of Range

Std True Range

Std Nav Range 

Fig. 2. Radiometric Standard Deviation Range to

target at Parachute Deploy

0 1 2 3 4 5 6 7 8 9 10

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Number of Monte Carlo Cases

Mean Range

Mean True Range

Mean Nav Range 

Fig. 3. Optical Mean Range to target at

Parachute Deploy

tical data shows so much less variability than the

radiometric data.

Figure 5 shows the 99.7% range for the two navigation types, again as a function of number of cases.

The dierence between the two navigation types is

even more pronounced for this statistic. The radiometric takes 50,000 cases to converge within 10% of

the nal result while the optical is within 2% from

2,000 cases on.In the runs using radiometric data,

the initial vehicle state is a much larger contributor

to the range than for the optical data. It has been

conjectured that the larger variation in the initial

state values may contribute to the slow convergence

of the radiometric data.

Not knowing the reason for the slow convergence, a conservative approach would require at least

60,000 cases to estimate the standard deviation of

2
0 1 2 3 4 5 6 7 8 9 10

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Number of Monte Carlo Cases

Standard Deviation of Range

Std True Range

Std Nav Range 

Fig. 4. Optical Standard Deviation Range to

target at Parachute Deploy

0 1 2 3 4 5 6 7 8 9 10

x 104

2.5

3

3.5

4

4.5

5

5.5

6

Number of Monte Carlo Cases

99.7% Range

Optical 

Radiometric

Fig. 5. 99.7% Range to target at Parachute

Deploy

the nal range.

4 Robustness Simulation

As a nal check, a 60,000 case Monte Carlo

simulation might be acceptable for a three degreeof-freedom (3DoF) problem. As a design tool, a

60,000 cases run is not practical with current computers. Fortunately a design tool need not estimate

the standard deviation or 99.7% range values. Since

the design is driven by the most severe cases, it is

only necessary to simulate those cases.

The current approach is \Monte Carlo-like" in

that it is based on random values for key simulation

parameters. The distributions of the random inputs

are modied so that the most extreme realistic cases

result. For the normally distributed variables, only

values more than 2 from the mean are used. For

uniform variables, only the endpoints of the distribution were used. In this manner, extreme (but

reasonable) combinations of parameters were investigated. These extreme values of inputs are referred

to as \robust" inputs.

Since the initial states were provided as-is and

not generated from a known distribution, it was necessary to select an appropriate set from the 100,000

states available. A set of 2,000 cases was chosen,

made up of ve distinct subsets. The rst subset

consisted of the 400 states with the largest position

delivery errors. The second subset consisted of the

400 states with the largest velocity delivery errors.

The third subset consisted of the 400 states with the

largest position knowledge errors. The fourth subset

consisted of the 400 states with the largest velocity

knowledge errors. The nal subset consisted of the

400 states with the largest sum of absolute value

of the error in each of the 12 state components (3

position, 3 velocity, 3 knowledge position, 3 knowledge velocity). That is, while the rst four subsets were vector magnitudes, the fth subset summed the

absolute value of each component of the other four vectors. This fth subset is referred to as the 12-

state magnitude error subset.

There is considerable correlation between the

ve subsets. From the radiometric data 68 states

occurred in only one subset, 447 states occurred in

exactly two subsets and 346 states occurred in exactly 3 subsets, for a total of 861 dierent states used

in the 2,000 runs. Figures 6 and 7 show the delivery

latitude and longitude of the 100,000 states and the

78.3 78.4 78.5 78.6 78.7 78.8 78.9

−11.9

−11.8

−11.7

−11.6

−11.5

−11.4

−11.3

True Longitude (deg)

True Latitude (deg)

3 km 10 km

Fig. 6. Initial Distribution of 100,000 radiometric

states and robust states.

861 selected. The endpoints of the 100,000 states

are included (usually multiple times) in the reduced

3
78.3 78.4 78.5 78.6 78.7 78.8 78.9

−11.9

−11.8

−11.7

−11.6

−11.5

−11.4

−11.3

True Longitude (deg)

True Latitude (deg)

3 km 10 km

Position Delivery Error 

Velocity Delivery Error 

Position Knowledge Error

Velocity Knowledge Error

12−State Mag Error Fig. 7. Initial Distribution of robust radiometric

states by subset.

set.

The states chosen from the optical data overlapped as well. They included 240 states in exactly

one subset, 532 states that occurred in exactly two

subsets, 229 in exactly three subsets, 1 in exactly

four subsets and 1 in exactly ve subsets, for a total

of 1,003 dierent states. Figures 8 and 9 show the

distribution of the 100,000 optical states and of the

78.45 78.5 78.55 78.6 78.65 78.7 78.75 78.8 −11.8

−11.75

−11.7

−11.65

−11.6

−11.55

−11.5

True Longitude (deg)

True Latitude (deg)

3 km 10 km

6 Fig. 8. Initial Distribution of 100,000 Optical

states and robust states.

1,003 selected initial states.

Figures 10 and 11 cover the navigated supersonic parachute deploy condition for the 100,000

case Monte Carlo simulation and the 2,000 case robustness study, respectively. Note that the simulation (exterior to the guidance) forced the parachute

to deploy whenever the navigated altitude dropped

below 6 kilometers, while the guidance attempted to

keep the parachute deployment within Mach and dy78.45 78.5 78.55 78.6 78.65 78.7 78.75 78.8 −11.8

−11.75

−11.7

−11.65

−11.6

−11.55

−11.5

True Longitude (deg)

True Latitude (deg)

3 km 10 km

Position Delivery Error 

Velocity Delivery Error 

Position Knowledge Error

Velocity Knowledge Error

12−State Mag Error 

7 Fig. 9. Initial Distribution of robust Optical states

by subset.

350 400 450 500 550 600 650 700 750 4

6

8

10

12

14

16

18

Nav Altitude (km)

Nav Relative Velocity (m/sec)

Fig. 10. Radiometric Navigated Conditions at

Supersonic Parachute Deploy; 100,000 Monte Carlo

cases.

namic pressure limits by constraining the parachute

deployment to occur at velocity between 370 and 503

m/s.. The footprint of the two studies are very similar. The robustness study resulted in more cases outside of the allowable Mach/Dynamic pressure box. This result is due to the more severe atmosphere in

the robustness runs, which was always at either its

thinnest or thickest value. Notice the banded structure in the Nav Altitude vs. Nav Velocity plot. This

banding will be even more evident in the optical navigation runs and will be discussed later. Figures 12

4
350 400 450 500 550 600 650 700 750 4

6

8

10

12

14

16

18

Nav Altitude (km)

Nav Relative Velocity (m/sec)

Fig. 11. Radiometric Navigated Conditions at

Supersonic Parachute Deploy; 2,000 Robust cases.

and 13 show the actual values to match the navigated values in gures 10 and 11.

350 400 450 500 550 600 650 700 750 4

6

8

10

12

14

16

18

True Altitude (km)

True Relative Velocity (m/sec)

Fig. 12. Radiometric True Conditions at

Supersonic Parachute Deploy; 100,000 Monte Carlo

cases.

Figures 14 and 15 are similar to the preceding

charts, but cover the optical states studies. Again

the results are very similar between the 100,000 cases

and the 2,000 cases, though the robustness cases

have more cases that exceed the parachute deploy

350 400 450 500 550 600 650 700 750 4

6

8

10

12

14

16

18

True Altitude (km)

True Relative Velocity (m/sec)

Fig. 13. Radiometric True Conditions at

Supersonic Parachute Deploy; 2,000 Robust cases.

350 400 450 500 550 600 650 700 750 4

6

8

10

12

14

16

18

Nav Altitude (km)

Nav Relative Velocity (m/sec)

Fig. 14. Optical Navigated Conditions at

Supersonic Parachute Deploy; 100,000 Monte Carlo

cases.

5
350 400 450 500 550 600 650 700 750 4

6

8

10

12

14

16

18

Nav Altitude (km)

Nav Relative Velocity (m/sec)

Fig. 15. Optical Navigated Conditions at

Supersonic Parachute Deploy; 2,000 Robust cases.

Mach limit. In gures 16 and 17, which show the

350 400 450 500 550 600 650 700 750 4

6

8

10

12

14

16

18

True Altitude (km)

True Relative Velocity (m/sec)

Fig. 16. Optical True Conditions at Supersonic

Parachute Deploy; 100,000 Monte Carlo cases.

true altitude and velocity at parachute deploy, the

banded structure is very distinct. The banding occurs because the altitude and velocity at parachute

deploy are largely determined by two factors: the atmospheric density and the hypersonic axial force coecient. For this mission the eect of the two factors

is approximately equal and in the robust case study,

350 400 450 500 550 600 650 700 750 4

6

8

10

12

14

16

18

True Altitude (km)

True Relative Velocity (m/sec)

Fig. 17. Optical True Conditions at Supersonic

Parachute Deploy; 2,000 Robust cases.

each factor was either at its maximum or its minimum. When a high density atmosphere combined

with a high axial force coecient, the parachute deploy was high and slow. When both factors were

low, the deploy was low and fast, and when one was

high and the other low, the result was in the middle. This eect is clear from gures 18 and 19 which

discriminate between high and low values for each

420 440 460 480 500 520 540 560 580 6

7

8

9

10

11

12

13

14

15

True Altitude (km)

True Relative Velocity (m/sec)

High CA (M<5)

Low CA (M<5) 

Supersonic Chute Deploy

EMQ Robust Cases − Radiometric + Optical − 02−Feb−2001

1 Fig. 18. Conditions at Parachute Deploy sorted

by Axial Force Coecient.

of the factors. Notice that all of the cases with high

atmospheric density and high axial force coecient

are on the upper left, all cases with low for both are

in the lower right and in the middle each case is low

in one and high in the other.

6
420 440 460 480 500 520 540 560 580 6

7

8

9

10

11

12

13

14

15

True Altitude (km)

True Relative Velocity (m/sec)

High Density Atmosphere

Low Density Atmosphere 

Supersonic Chute Deploy

EMQ Robust Cases − Radiometric + Optical − 02−Feb−2001

2 Fig. 19. Conditions at Parachute Deploy sorted

by Atmospheric Density. Prior to the Monte Carlo study, success criteria were established to determine what constituted a

\safe" landing. The maximum safe landing vertical velocity was 4 m/s and the maximum safe landing

horizontal velocity was 2 m/s. According to these

criteria, 180 cases in radiometric and 175 cases in optical failed the 100,000 case Monte Carlo, while 199

cases in radiometric and 132 cases in optical failed

the robustness study. 5 Postmortem and Future

The means of state selection described above was considered ad hoc. In an eort to make the state

selection process better, the footprint and parachute

deploy conditions for the robustness study were broken into subsets corresponding to the initial condition subsets. The segregated footprint is shown

in gure 20. It is clear that all of the very long

cases and most of the very short cases resulted from

states in the 12-state magnitude subset. Figure 21

shows the parachute deploy conditions similarly segregated. Here all ve subsets contribute to the

out-of-bounds cases which are driven largely by the

atmosphere. But the 12-state magnitude states seem

to cover the range of out-of-bounds cases. Figures 22

and 23 show the same results for the optical cases.

In the footprint shown in gure 22, there is more

of a mixture of subsets in the very long and very

short cases, but most seem to be from knowledge

errors and 12-state errors. This result is due to the much smaller state errors for the optical cases, so the

knowledge errors become the dominant factor and

the 12-state errors include the knowledge errors. Finally, gure 23 shows the deploy condition for the

88.5 89 89.5 90 90.5 91 91.5

3.5

4

4.5

5

5.5

6

True Longitude (deg)

True Latitude (deg)

3 km

Position Delivery Error 

Velocity Delivery Error 

Position Knowledge Error

Velocity Knowledge Error

12−State Mag Error Fig. 20. Radiometric Footprint broken down by

initial state subset.

400 500 600 700 800 900 1000 1100 1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

Dynamic Pressure (N/m2)

Mach Number

Position Delivery Error 

Velocity Delivery Error 

Position Knowledge Error

Velocity Knowledge Error

12−State Mag Error Fig. 21. Radiometric Parachute Deploy

Conditions broken down by initial state subset.

optical cases. Here almost all of the out-of-bounds

cases are either knowledge or 12-state.

Figures 20 through 23 show that the 12-state

magnitude errors show up consistently in the extreme cases. Suggesting that this method may be

a way to select states for future robustness studies,

perhaps with some weighting of the various components to nd the most eective states.

Another issue is whether the endpoint of a uniform distribution is the worst possible input when

it is combined with other inputs. This issue is espe7
90.15 90.2 90.25 90.3 90.35 90.4 90.45

5.35

5.4

5.45

5.5

5.55

True Longitude (deg)

True Latitude (deg)

3 km

Position Delivery Error 

Velocity Delivery Error 

Position Knowledge Error

Velocity Knowledge Error

12−State Mag Error Fig. 22. Optical Footprint broken down by initial

state subset.

400 500 600 700 800 900 1000 1100 1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

Dynamic Pressure (N/m2)

Mach Number

Position Delivery Error 

Velocity Delivery Error 

Position Knowledge Error

Velocity Knowledge Error

12−State Mag Error Fig. 23. Optical Parachute Deploy Conditions

broken down by initial state subset.

cially a concern for the atmosphere where a very

complicated, non-linear system has been parameterized by a single factor, the atmospheric opacity. In the future studies we will replace the endpoints

of the uniform variables with \V"-distributed variables. That is, variables that have 0 probability of

occurring at the midpoint and linearly increase in

probability away from the midpoint.

6 Conclusions

The number of runs needed for a full Monte

Carlo analysis has been found to be much too large

for its use as a design tool for planetary entry problems. A \Monte Carlo-like" design analysis tool has

been developed and applied to a Mars entry vehicle. This tool provides realistic but challenging design cases using many fewer cases than a full Monte

Carlo analysis. The problem of random input variables that are provided a priori (as opposed to being

drawn from a given distribution) has been addressed

and a solution was found that shows prospects of future improvement.

References

1Braun, R.D., Powell, R.W., Engelund, W.C.,

Gnoo, P.A., Weilmuenster, K.J., Mitcheltree, R.A., \Mars

Pathnder Six-Degree-of-Freedom Entry Analysis," Journal of Spacecraft and Rockets,Vol.32, No.6, pp.993-1000,

November-December, 1995. 2Justus,C.G., \Mars Global Reference Atmospheric

Model for Mission Planning and Analysis," Journal of Space- craft and Rockets,Vol.28, No.2, pp. 216-221, March-April,

1991. 3Bauer, G.L., Cornick, D.E. and Stevenson, T. \Capabilities and Applications of the Program to Optimize Simulated Tra jectories (POST)," NASA CR-2770, February, 1977.

8
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A Approach to Simulation of Extreme Conditions for a Planetary Lander

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Eric M. Queen, Scott A. Striepe, Richard W. Powell 

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13. ABSTRACT (Maximum 200 words)

A ``Monte Carlo-like" design analysis tool is developed and applied to an aeromaneuvering Mars entry vehicle.

This tool provides realistic but challenging design cases using many fewer cases than a full Monte Carlo

analysis. The problem of random input variables that are provided a priori (as opposed to being drawn from a

given distribution) is addressed and a solution is found that shows prospects for future improvement.

14. SUBJECT TERMS

Monte Carlo Design; Mars Lander; Trajectory Analysis

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