Rafael Lugo, Robert Tolson Department of Mechanical and Aerospace Engineering

1. Entry, Descent, and Landing Trajectory and Atmosphere Reconstruction with Uncertainty Quantification using Monte Carlo Techniques Rafael Lugo, Robert Tolson Department of Mechanical and Aerospace Engineering North Carolina State University, Raleigh, NC and Robert Blanchard National Institute of Aerospace, Hampton, VA 10th International Planetary Probe WorkshopSan Jose State University20 June 2013

2. 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

3. 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) Image courtesy of JPL

4. MSL Entry, Descent, and Landing Rover Separation Body accelerations & rates from IMU Sky Crane Detail Initial state from radio tracking & star tracker Mobility Deploy Entry Interface Flyaway Touchdown Landing Location Hypersonic Aero-maneuveringBegins Parachute Deploy Peak Heating Backshell Separation Peak Deceleration Heatshield Separation Pressures & temps from MEDLI Radar based solution converged Powered Descent vertical flight Sky Crane (see inset) Radar Altimeter Flyaway Images courtesy of JPL & Honeywell

5. 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

6. Classical Reconstruction Initial conditions IMU error parameters Accelerations Angular rates Inertial navigation Wind anglesα, β Vehicle stateR,V, q ax V h Axial force equation CFD force coefficient Freestream densityρ Gravity acceleration Hydrostatic equation Freestream pressureP NO redundant data NO uncertainties M = molar mass R = gas constant Equation of state Freestream temp.T

7. INSTAR • How can we utilize redundant data and obtain meaningful statistics from inertial navigation? • Answer: Utilize Monte Carlo techniques • Disperse initial state conditions and IMU error parameters (acceleration & rate biases, scale factors, & misalignments) with specified uncertainties (covariance) • Integrate IMU data (inertial navigation) using these dispersed initial conditions to obtain set of dispersed trajectories • Obtain mean initial conditions and statistics (covariance) from subset of trajectories that satisfy redundant observations to within specified tolerances • 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

8. 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 Measurement space Landing location Initialconditions Statistical analysis of Altimetry IMU error parameters Representative of true and unknowable ICs, IMU parameters, and uncertainties FADSpressures

9. 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 • 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

10. 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 Δsare under 1σ • Exceptions: 𝜃Y& z-axis bias • Results are used to begin next step: inclusion of FADS data Parameter changes from a priori

11. 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 Dispersed trajectories Atmosphere Reconstruction A priori P0, M0 Dispersed atmospheres CFD Transducer errors Dispersed model pressures Monte Carlo distributions Trajectory downselection Updated ICs, covariance, transducer errors FADS observations Updated trajectory & atmosphere INSTAR Updated uncertainties

12. MEADS Observations

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

14. 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

15. 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

16. 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)

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

18. 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 1σuncertainties 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 solutions Acknowledgements • 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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20. Backup Slides

21. 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) Landing site location

22. 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 process • Disperse initial conditions and IMU error parameters with specified uncertainties • Integrate IMU data using these initial conditions to obtain set of dispersed trajectories • Obtain statistics and mean initial conditions from subset of trajectories that satisfy redundant data • Repeat until convergence Initial Conditions, MME@t0 frame