NC STATE UNIVERSITY

Rafael Lugo, Robert TolsonDepartment of Mechanical and Aerospace Engineering

North Carolina State University, Raleigh, NC

and

Robert BlanchardNational Institute of Aerospace, Hampton, VA

Entry, Descent, and Landing Trajectory and Atmosphere Reconstruction with Uncertainty Quantification using

Monte Carlo Techniques

10th International Planetary Probe WorkshopSan Jose State University

20 June 2013

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Introduction• Trajectory reconstruction – process by which vehicle

position, velocity, and orientation is determined post-flight

• Goals of post-flight trajectory reconstructions– Validate pre-flight models– Aid in planning of future EDL missions by identifying areas of

improvement in vehicle design, e.g., reduce mass of heat shield because of overly conservative pre-flight aeroheating models

• Mars exploration: Missions with EDL operations– Viking 1 & 2 (20 July & 3 September 1976)– Pathfinder (4 July 1997)– Mars Exploration Rover A & B (4 & 25 January 2004)– Phoenix (25 May 2008)– Mars Science Laboratory (6 August 2012)– InSight (2016), MSL 2020

• Redundant sensors (e.g., sensors other than accelerometers and rate gyroscopes) enable better trajectory estimates

Image courtesy of JPL

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Mars Science Laboratory• Landed in Gale Crater on 6 August 2012• Largest and most sophisticated Mars vehicle

– Required innovative landing technique: “Skycrane” landing used hovering platform to lower rover to surface using umbilicals

– Guided entry (first for Mars)• Entry vehicle equipped with two IMUs, guidance and

navigation computer, and Mars EDL Instrumentation (MEDLI) suite – MEDLI included series of 7 pressure ports on heat shield

that measured pressures during entry and descent– Pressure ports formed the Mars Entry Atmospheric Data

System (MEADS)Viking Pathfinder MER Phoenix MSL

Diameter (m) 3.505 2.65 2.65 2.65 4.518

Entry mass (kg) 930 585 840 602 3152

m/CDA 63.7 62.3 89.8 65 135

Hypersonic L/D 0.18 0 0 0 0.24

Hypersonic αtrim (deg) 11.2 0 0 0 16

Control 3-axis, unguided Spinning Spinning Uncontrolled 3-axis, guidedImage courtesy of JPL

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MSL Entry, Descent, and Landing

Flyaway

Hypersonic Aero-maneuvering Begins

Entry Interface

Peak Deceleration

Parachute Deploy

Heatshield SeparationRadar based solution

converged

Backshell Separation

Powered Descent vertical flight

Sky Crane (see inset)

Rover Separation

Mobility Deploy

Touchdown

Flyaway

Sky Crane Detail

Peak Heating

Pressures & temps from MEDLI

Radar Altimeter

Landing Location

Body accelerations & rates from IMU

Initial state from radio tracking & star tracker

Images courtesy of JPL & Honeywell

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Trajectory Reconstruction• Generally, trajectory parameters are not measured directly and must be either estimated

probabilistically or computed deterministically• Two common techniques

– Probabilistic/statistical approach – measurements processed in a stochastic algorithm that minimizes payoff function defined by system models (e.g., WLS, MV, EKF, UKF, etc.)

– Inertial navigation – deterministic technique where acceleration and attitude rate measurements from an inertial measurement unit (IMU) are integrated given an initial state

• If IMU measurements and IC estimates were perfect, inertial navigation solution would be the true and unknowable trajectory in inertial space

Method Advantages DisadvantagesStatistical methods • Can blend redundant data types

(pressures, altimetry, etc.)• Parameter statistics readily

available

• Requires dynamical models (atmosphere, aerodynamics, data & state noise, observation equations, etc.)

• Often requires filter “tuning” to assure convergence

Inertial navigation • Only gravity model required• No convergence issues

• Cannot include redundant measurements

• Uncertainties not readily available

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Classical Reconstruction

Accelerations Inertial navigation

Vehicle stateR, V, q

CFD force coefficient

Freestream densityρ

Freestream pressureP

Freestream temp.T

Wind anglesα, β

Hydrostatic equationGravity acceleration

h

ax

V

Equation of state

Angular rates

Initial conditions

P g h

ΜPTR

22 , x

A AA

ma C CV C S

M = molar massR = gas constant

NO redundant dataNO uncertainties

IMU error parameters

Axial force equation

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INSTAR• How can we utilize redundant data and obtain meaningful statistics from inertial

navigation?• Answer: Utilize Monte Carlo techniques

1. Disperse initial state conditions and IMU error parameters (acceleration & rate biases, scale factors, & misalignments) with specified uncertainties (covariance)

2. Integrate IMU data (inertial navigation) using these dispersed initial conditions to obtain set of dispersed trajectories

3. Obtain mean initial conditions and statistics (covariance) from subset of trajectories that satisfy redundant observations to within specified tolerances

4. With this new set of initial conditions and covariance, repeat steps 1-3 until convergence• Inertial Navigation Statistical Trajectory and Atmosphere Reconstruction (INSTAR)

– “Debuted” at 2013 AAS/AIAA Spaceflight Mechanics Conference– Demonstrated INSTAR using landing site location as redundant data

Integrator• INSTAR integrator is a fixed-step, three-point predictor-corrector• Integrator written in Fortran95 with a MATLAB wrapper

– Utilizes multi-core processing to integrate multiple trajectories simultaneously– Accelerations and rates for 1,000 trajectories (from EI-9 min to landing) can be integrated in <3 min

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INSTAR Overview• Trajectory state can be mapped using inertial navigation into “measurement space” where

redundant observations and uncertainties can be introduced• Subset of trajectories that satisfy redundant data provide updated initial conditions, IMU

error parameters, and covariance

Initialconditions

IMU error parameters

Statisticalanalysis of

Landing location

FADSpressures

Altimetry

Measurement space

Representative of true and unknowable ICs, IMU parameters, and

uncertainties

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INSTAR Results: IC & IMU Dispersions• Dispersed initial state to within given uncertainties, 10,000 cases (Gaussian distribution),• 26 valid trajectories within 150 m of reference landing site (uniform distribution)• Mean of ICs and full covariance computed from 26 valid trajectories, use these new initial

conditions and new covariance to disperse new set of trajectories

-8000 -6000 -4000 -2000 0 2000 4000 6000-8000

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adiu

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Dispersions (10000 cases)Valid (26 cases)ReferenceNominal

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Dispersions (5000 cases)Valid (3855 cases)ReferenceNominal

-400 -300 -200 -100 0 100 200 300 400 Areocentric latitude [m]

• Significantly smaller distribution area• Select trajectories that land within 150 m

of landing site in latitude-longitude-radius space

• Mean of ICs and full covariance computed from valid trajectories

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INSTAR Results – Landing Location• Continue to iterate, results

are from 6th iteration• Result: updated ICs and

uncertainties, incorporating redundant data (landing site location), using only inertial navigation– Improved landing site

location difference from 925 m to 19 m

• Nearly all uncertainties have decreased from a priori

• Nearly all IC Δs are under 1σ– Exceptions: 𝜃Y & z-axis bias

• Results are used to begin next step: inclusion of FADS data

Initial Conditions 1σ UncertaintyParameter Units Δ Final Value Δ Final Value

X m -0.7467 -8.969413E+04 0.6925 6.743Y m -2.4342 5.080897E+06 -0.4728 6.117Z m -5.4462 -9.913041E+04 -0.7378 17.272VX m/s -0.0003 -3.983226E+03 -0.0049 0.005VY m/s -0.0015 -3.685552E+03 -0.0021 0.008VZ m/s 0.0001 -2.792489E+02 -0.0009 0.009𝜃X deg 0.0110 -156.1207 -0.0120 0.018𝜃Y deg 0.0266 -65.9161 -0.0227 0.007𝜃Z deg 0.0080 -157.6911 -0.0090 0.021Ba,x μg 1.0609 1.0609 5.2581 38.591Ba,y μg -15.7123 -15.7123 -9.6127 23.721Ba,z μg -37.3436 -37.3436 -21.9187 11.415

Parameter changes from a priori

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, , , , , : ,i t i iP f q M P P

FADS Data in INSTAR• Flush air data systems measure surface pressures Pi at port locations defined by (ηi , ζi)– Statistical methods (e.g., least squares) may be used

to obtain aerodynamic and atmospheric parameters • Process: Given dispersed trajectories from

INSTAR, compute CFD pressures & disperse using transducer scale factors and biases, and compare them to observed MEADS pressures– This method is be comparable to how INSTAR works

with other redundant data– CFD errors affect solution, since model pressures and

CA are obtained from CFD tables

Updated uncertainties , , , , , , , ,R V q M P T

Dispersed trajectories, , , , R V q

Dispersed atmospheres , , , ,h M P T

INSTAR

FADS observations

Updated trajectory & atmosphere , , , , , , , ,M P T R V q

Transducer errors

Monte Carlo distributions

Atmosphere Reconstruction

A priori P0, M0

CFD

Dispersed model pressures

Updated ICs, covariance,

transducer errors

Trajectory downselection

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MEADS Observations

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Atmosphere Dispersions• Atmosphere reconstruction is

performed for each of 1,000 dispersed trajectories

• For each atmosphere profile, initial pressure is taken from model that consists of two averaged mesoscale models anchored to surface pressure measurements from the Curiosity rover

• Behavior at 13 km corresponds to region of trajectory where altitude increases

P g h

ΜPTR

22 , x

A AA

ma C CV C S

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Pressure Dispersions• Mach number histories are

computed and used with wind angle histories to look up model pressures from pre-flight CFD database

• Result is a set of 7x1000 CFD-based pressure histories that are further dispersed using randomized biases and scale factors– Normal distribution, 25 Pa bias, 0.02

scale factor• Dispersed model pressures are

compared to the MEADS observations– Residuals from other ports display

comparable behavior– Black curve: difference between

nominal pressures & observations

CFD , ,P f M

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Pressure Dispersions - Downselected• Dispersed trajectories are down-

selected by choosing those trajectories with residuals that are within a priori MEADS measurement uncertainties

• Result is a subset of 33 “valid” trajectories that satisfy a priori pressure measurement uncertainties, which are a subset of original 1,000

• Note decrease in magnitudes of residuals from previous slide

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Results – Updated Initial Conditions• Compute means and standard deviations of initial conditions of 33 valid trajectories, repeat

INSTAR process on to 4th iteration• Nine of twelve state parameter uncertainties have further decreased (slightly) from solution

obtained using only landing site location– All Δref are below 1σ, except for 𝜃Y and Ba,z

– Improvements to initial state are limited because landing site already refined these• New landing site is now 16.3 m away from reference (compare to 18.9 m)

Initial Conditions 1σ UncertaintyParameter Units Δref Δnom Value Δref Δnom Value

X m -0.4038 0.3429 -8.969378E+04 0.6684 -0.0241 6.718Y m -2.6699 -0.2357 5.080896E+06 -0.3017 0.1711 6.288Z m -5.8695 -0.4233 -9.913083E+04 -0.7489 -0.0111 17.261VX m/s -0.0005 -0.0002 -3.983227E+03 -0.0051 -0.0002 0.005VY m/s -0.0008 0.0007 -3.685551E+03 -0.0008 0.0013 0.009VZ m/s 0.0003 0.0003 -2.792487E+02 -0.0012 -0.0003 0.009𝜃X deg 0.0129 0.0019 -156.1187 -0.0162 -0.0042 0.014𝜃Y deg 0.0262 -0.0003 -65.9164 -0.0239 -0.0013 0.006𝜃Z deg 0.0055 -0.0026 -157.6936 -0.0139 -0.0049 0.016Ba,x μg 4.1872 3.1263 4.1872 0.1782 -5.0799 33.512Ba,y μg -19.2161 -3.5038 -19.2161 -11.1737 -1.5610 22.160Ba,z μg -38.6221 -1.2785 -38.6221 -21.1398 0.7789 12.194

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Atmo Uncertainties & Transducer Errors • Transducer biases and scale factors

that correspond to valid trajectories are averaged to obtain the new set of transducer biases and scale factors, as in INSTAR process– Uncertainties obtained by computing

standard deviations of this subset– Uncertainties have improved from a

priori values (25 Pa bias, 0.02 scale factor)

Bias (Pa) Scale FactorPort Value 1σ Value 1σ

1 9.1665 16.8264 1.0117 0.00342 5.0794 22.1129 1.0082 0.00473 16.7960 25.3761 1.0091 0.01074 16.7052 20.6148 1.0102 0.00455 7.4363 20.3960 1.0011 0.00976 22.1641 16.9234 1.0045 0.00607 10.4546 14.2642 1.0083 0.0056

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m3 )

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• Recall that uncertainties are due to trajectory IC dispersions, which are small– CFD errors are not considered (yet)

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Summary & Conclusions• Uncertainties and updated EDL trajectory ICs can be obtained using inertial navigation and

Monte Carlo dispersion techniques• Demonstrated INSTAR using MSL EDL data

– Redundant data: Landing site location, MEADS pressures• Obtained updated pressure transducer biases and scale factors, initial conditions, and

associated uncertainties– Significantly improved landing site location by adjusting initial state– Including pressures provides transducer errors but requires use of CFD models for pressures and CA

• Final IC & acceleration bias Δs were well under 1σ uncertainties for nearly all parameters – Exceptions: Initial Y-axis Euler angle (4.3σ) and Z-axis acceleration bias (3.2σ)

Work in Progress• Account for CFD errors in dispersions to get more realistic atmosphere uncertainties• Compare transducer errors to those computed using FADS-based statistical solutionsAcknowledgements• NASA Langley: Mark Schoenenberger, Chris Karlgaard, Prasad Kutty, Jeremy Shidner, David

Way, Chris Kuhl, Michelle Munk• JPL MSL EDL & navigation teams

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References1. Crassidis, J. L., and Junkins, J. L., Optimal Estimation of Dynamic Systems, Chapman & Hall/CRC, Washington, D.C., 2004, Chaps. 1, 2.2. Grotzinger, J. P., Crisp, J., Vasavada, A. R., Anderson, R. C., Baker, C. J., Barry, R., Blake, D. F., Conrad, P., Edgett, K. S., Ferdowski, B., Gellert, R., Gilbert, J.

B., Golombek, M., Gomez-Elvira, J., Hassler, D. M., Jandura, L., Litvak, M., Mahaffy, P., Maki, J., Meyer, M., Malin, M. C., Mitrofanov, I., Simmonds, J. J., Vaniman, D., Welch, R. V., and Wiens, R. C., “Mars Science Laboratory Mission and Science Investigation,” Space Science Reviews, Vol. 170, 2012, pp. 5–56.

3. Karlgaard, C. D., Kutty, P., Schoenenberger, M., Shidner, J., and Munk, M., “Mars Entry Atmospheric Data System Trajectory Reconstruction Algorithms and Flight Results,” 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Grapevine, TX, January 2013, AIAA 2013-0028.

4. Way, D. W., Powell, R. W., Chen, A., Steltzner, A. D., San Martin, A. M., Burkhart, P. D., Mendeck, G. F., “Mars Science Laboratory: Entry, Descent, and Landing System Performance,” IEEE 2006 Aerospace Conference, March 2006.

5. Striepe, S. A., Way, D. W., Dwyer, A. M., and Balaram, J., “Mars Science Laboratory Simulations for Entry, Descent, and Landing,” JSR, Vol. 43, No. 2 (2006), pp. 311-323.

6. Blanchard, R. C., Tolson, R. H., Lugo, R. A., Huh, L., “Inertial Navigation Entry, Descent, and Landing Reconstruction using Monte Carlo Techniques,” 23rd AAS/AIAA Spaceflight Mechanics Meeting, Kauai, HI, February 2013, AAS 13-308.

7. Karlgaard, C. D., Kutty, P., Schoenenberger, M., and Shidner, J., “Mars Science Laboratory Entry, Descent, and Landing, Trajectory and Atmosphere Reconstruction,” 23rd AAS/AIAA Spaceflight Mechanics Meeting, Kauai, HI, February 2013, AAS 13-307.

8. Gazarik, M. J.,Wright, M. J., Little, A., Cheatwood, F. M., Herath, J. A., Munk, M. M., Novak, F. J., and Martinez, E. R., “Overview of the MEDLI Project,” IEEE 2008 Aerospace Conference, March 2008.

9. Munk, M., Hutchinson, M., Mitchell, M., Parker, P., Little, A., Herath, J., Bruce, W., and Cheatwood, N., “Mars Entry Atmospheric Data System (MEADS): Requirements and Design for Mars Science Laboratory (MSL),” 6th International Planetary Probe Workshop, Atlanta, GA, June 2008.

10. Blanchard, R.C., Desai, P.N., “Mars Phoenix Entry, Descent, and Landing Trajectory and Atmosphere Reconstruction,” Journal of Spacecraft and Rockets, Vol. 48, No. 5, 2011, pp. 809-21.

11. Dyakonov, A., Schoenenberger, M., and Van Norman, J., “Hypersonic and Supersonic Static Aerodynamics of Mars Science Laboratory Entry Vehicle,” 43rd AIAA Thermophysics Conference, New Orleans, LA, June 2012, AIAA 2012-2999.

12. Pruett, C. D., Wolf, H., Heck, M. L., and Siemers, P. M., “Innovative Air Data System for the Space Shuttle Orbiter,” Journal of Spacecraft and Rockets, Vol. 20, No. 1, 1983, pp. 61-69.

13. Siemers III, P. M., Henry, M. W., and Flanagan, P.F ., “Shuttle Entry Air Data System Concepts Applied to Space Shuttle Orbiter Flight Pressure Data to Determine Air Data - STS 1-4,” 21st AIAA Aerospace Sciences Meeting, Reno, NV, January 1983, AIAA 83-018.

14. Vasavada, A. R., Chen, A., Barnes, J. R., Burkhart, P. D., Cantor, B. A., Dwyer-Cianciolo, A. M., Fergason, R. L., Hinson, D. P., Justh, H. L., Kass, D. M., Lewis, S. R., Mischna, M. A., Murphy, J. R., Rafkin, S. C. R., Tyler, D., and Withers, P. G., “Assessment of Environments for Mars Science Laboratory Entry, Descent, and Surface Operations,” Space Science Reviews, Vol. 170, 2012, pp. 793–835.

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Backup Slides

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Coordinate Frames• Trajectory integrated in Mars-centered, Mars

mean equator (MME) frame fixed at the Prime Meridian date of t0 (“M frame”)

• Utilized IMU observations in descent stage (DS) frame, transformed to inertial M frame

• Wind angles computed from body velocity components

• Landing site location supplied by JPL– Assumed to be accurate to within 150 m in

longitude, latitude, and radius (i.e., uniform distribution)

Parameter Units ValueEast longitude deg 137.4417Areocentric latitude deg -4.5895Radius m 3391133.3Radius + tether length (9.4 m) m 3391142.7

Landing site location

MZ

MY

MX

Mars Mean Equator at 0t

Prime Meridian at 0tNorth Pole

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INSTAR Trajectory Reconstruction• Initial time: t0 – 10s

– t0 defined to be 9 min prior to entry interface

• Initial state (position, velocity, orientation) and covariance at t0 – 10s provided by JPL– Solving for 12 parameters (9 initial conditions

& 3 IMU errors)• Gravity model: central + J2

• Used acceleration and rate data without smoothing or filtering

• Recall INSTAR process1. Disperse initial conditions and IMU error

parameters with specified uncertainties2. Integrate IMU data using these initial

conditions to obtain set of dispersed trajectories

3. Obtain statistics and mean initial conditions from subset of trajectories that satisfy redundant data

4. Repeat until convergence

Parameter Units Value 1σX m -8.969338E+04 6.05Y m 5.080899E+06 6.59Z m -9.912496E+04 18.01VX m/s -3.983226E+03 0.01VY m/s -3.685550E+03 0.01VZ m/s -2.792490E+02 0.01𝜃X deg -156.131612 0.03𝜃Y deg -65.942677 0.03𝜃Z deg -157.699118 0.03Ba,x μg 0 33.3Ba,y μg 0 33.3Ba,z μg 0 33.3

Initial Conditions, MME@t0 frame