Richard Slade, EADS Astrium, Stevenage, UK

1. 11th AIAA / ISSMO MAO Conference, 6-9 September 2006, Portsmouth, VA Analysis, Optimization and Probabilistic Assessment of an Airbag Landing System for the ExoMars Space Mission Richard Slade, EADS Astrium, Stevenage, UK Royston Jones and Paul Sharp, Altair Engineering Ltd, Coventry, UK and Vassili Toropov, University of Leeds (with Altair Engineering until April 2006), UK

2. ExoMars Space Mission • ESA Aurora exploration programme • 240kg mobile robotic exo-biology laboratory • To search for extinct or extant microbial life on Mars • Supporting geology and meteorology experiments • Launch in 2011 or 2013 • Currently in Phase B – mission planning and concept design phase

3. Airbags for Space Landers • Un-vented Type • Multiple bounces • Established Heritage • High Mass • Vulnerable to rupture • Vented Type • Active control • Single stroke • No space heritage • Low Mass • Vulnerable to over-turning Mars Pathfinder NASA/JPL Beagle 2 Beagle2 Kistler Booster Irvin ExoMars ESA

4. Airbag Landing Design Concept • Design concept considers vented (or “Dead-Beat”) airbag coming to rest on second bounce • Inflated with N2 during descent under main parachute

5. Airbag Configuration • Six identical vented chambers • One “anti-bottoming” un-vented toroidal

6. Optimization and reliability assessment of ExoMars lander Failure modes: • Roll-over (payload overturns), • Dive-through (payload impacts rock) • Rupture (fabric tears) • Full-scale terrestrial testing is difficult / expensive

7. LS-DYNA Dynamic Relaxation (steady-state free fall condition) Airbag functionality (Wang Nefske inflation model) Advanced contact (internal fabric contact etc.) HyperMesh Advanced LS-DYNA model build support Comprehensive interface HyperView Time dependent LS-DYNA animations Multi results type environment (animations, X-Y data) HyperMorph Airbag parameterisation Rock height, pitch angle variation in reliability assessment HyperStudy Airbag size optimisation Reliability assessment Software Tools

8. Landing Scenarios Two Landing Scenarios – Flat Bottom & Inclined Rock Impacts Mars Environment: • Gravity 3.7 m/s2 = 0.38g • Pressure 440Pa = 0.4% of Earth air pressure at sea level • = at 36.5 km altitude on Earth • Temperature 187K = - 86º C

9. Landing Scenarios Landing scenarios are chosen to give conflicting design requirements • Flat Bottomed Impact • Vertical Velocity 25m/s • Favours ‘Tall’ airbag designs • Favours ‘Narrow’ airbag designs • Tall, Narrow airbag makes most effective vertical energy absorber • Inclined Rock Impact • Vertical Velocity 25m/s • Lateral ‘Wind’ Velocity 16.3m/s • Favours ‘Wide’ Airbag designs • Wide airbag makes most effective rock intrusion absorber

10. Baseline Response : Flat Bottom Impact • Peak Filtered Deceleration 66g (Target <70g) • Peak Airbag Material Stresses 135MPa (Target <533MPa) • Constraints Satisfied by Baseline

11. Baseline Response : Inclined Rock Impact • Peak Filtered Deceleration 980g (Target <70g • Peak Filtered Deceleration 980g (Target <70g) • Peak Airbag Material Stresses 281MPa (Target <533MPa) • Deceleration Constraint Exceeded Due to ‘Dive Through’

12. Baseline Response : Inclined Rock Impact ‘Dive Through’ • Critical to prevent ‘direct’ payload to Rock/Ground impact • Such type of impact guarantees violation of deceleration constraint Direct Payload to Rock impact due to ‘dive through’

13. ExoMars Lander: LS-DYNA Simulation

14. Optimization: Set-up • Design Objective- Minimise System Mass (Airbag + Payload + Gas + System) • Design Constraints- Payload Acceleration (<70g) • - Global Bag Von Mises Stress (<533MPa) - Re-bound and “roll over” inversion kinematics Design Objective and Constraint Responses Evaluated for each Landing Case Design Variables- Airbag Base Diameter (HyperMorph) - Airbag Height (HyperMorph) - Airbag Venting Area - Airbag Steady-State Pressure (Mass of Gas) • Minimise Design Objective by varying the Design Variables whilst satisfying the Design Constraints

15. Design Parameterization ¼ ellipse ¼ ellipse ¼ ellipse Design Variables : Airbag Height and Diameter • Airbag geometry defined by dimensional relationships between height (H) and diameter (D) of cross-section, curves are elliptical sections • Geometric factors a, b, c are constant

16. Metamodelling • Need for metamodelling • One LS DYNA analysis of 0.2s after touchdown takes 10 hours of computing • Unifying approach • Both optimization and reliability study utilise metamodels • Accuracy of metamodels • Optimization and reliability studies based on metamodels • High quality metamodel is required

17. DOEs for metamodel building • Main requirements to DOE are: • maximum quantity of information • achieved with minimum computational effort (number of numerical experiments)

18. Optimal Latin Hypercube DOEs • Optimal Latin Hypercube (OLH) DOEs specify the sample points such that as much of the design space is sampled as possible, with the minimum number of response evaluations - especially useful when the evaluations are expensive. • OLH DOEs are highly structured in a way that: • They provide an optimal uniform distribution of sample points. • They spread out the sample points efficiently (space filling) through out the design space. • OLH can also be used to specify sampling points in robust design, reliability analysis.

19. LH DOE – conventional (random) and optimal Random Latin hypercube Optimal Latin hypercube random points distribution optimal uniformity of the points distribution

20. Optimization: Objective function (Audze-Eglais 1977) min = min • Physical analogy: a system consisting of points of unit mass exert repulsive forces on each other causing the system to have potential energy. When the points are released from an initial state, they move. They will reach equilibrium when the potential energy of the repulsive forces between the masses is at a minimum. If the magnitude of the repulsive forces is inversely proportional to the distance squared between the points then minimize: Potential energy (objective function) Distance between points p and q

21. permGA iteration history for 2 design variables & 10 points

22. permGA iteration history for 2 design variables & 50 points

23. permGA iteration history for 2 design variables & 400 points

24. permGA iteration history for 2 design variables & 999 points

25. Metamodelling: DoE Four Design Variables – 40 Test Plan Points (EULH) / Landing Scenario

26. Metamodel building using Moving Least Squares Method (MLSM) • Suggested for generation of surfaces given by points • Used in meshless (mesh-free) form of FEM • Useful for metamodel building • Simple – can be explained to (and understood by) an engineer within 5 minutes

27. Moving Least Squares Method Generalization of a weighted least squares method where weights do not remain constant but are functions of Euclidian distance rk from a k-th sampling point to a point x where the surrogate model is evaluated. DoE point Evaluation point x xj rk xi

28. Moving Least Squares Method The weight wi , associated with a sampling point xi , decays as a point x moves away from xi . Because the weights wi are functions of x, the polynomial basis function coefficients are also dependent on x. This means that it is not possible to obtain an analytical form of the approximation function but its evaluation is still computationally inexpensive.

29. Gaussian weight decay function wi= exp(-ri2) where  is “closeness of fit parameter”  = 1  = 10  = 100

30. Example: six-hump camel back function F(x1,x2) = (4 - 2.1 x12 + x14 / 3) x12+x1x2+(- 4 + 4x22) x22, -2 ≤ x1 ≤ 2, -1 ≤ x2 ≤ 1.

31. Example: six-hump camel back function Original function Approximation on 100 sampling points,  = 120

32. Example: Computing and Rendering Point Set Surfaces by M. Alexa et al. 2001 larger  smaller 

33. Metamodel Generation using MLSM • MLSM provides a high quality response surface to accurately approximate a highly nonlinear system. • Important feature of MLSM is efficient handling of numerical noise by adjusting “closeness of fit” parameter to provide close fit to a low noise situation or loose fit when the response exhibits a larger amount of noise • Direct Payload to Rock/Ground Impact Resulting in High Payload Accelerations (>100g) occurred at high percentage of Test Plan Points • These high results ‘swamp’ the responses of interest in the approximation

34. MLSM Test Example : Rosenbrock’s Banana Valley Function Function Capped Noise Outside Area of Interest • To minimise this function, a good quality approximation of the valley should be obtained whilst ignoring numerical noise outside the valley

35. MLSM Test Example : Rosenbrock’s Banana Valley Function Least Squares approximation of capped function, 100 Sampling points, quadratic polynomial (left) and cubic polynomial (right), still give poor approximation of function

36. MLSM Test Example : Rosenbrock’s Banana Valley Function MLSM approximation of capped function, 100 sampling points gives good approximation of function Capped function

37. Optimization results • On review of the response data set obtained from the Test Plan points it was observed that there was a high percentage of runs that failed to meet the constraints • This was reflected in the approximations and resulted in a very small ‘sweet spot’ on the response surface where the constraints could be met • Model mass varied very little (<0.25%) within this area (all runs in this area had more mass than baseline run) • Because of the minimal mass penalty, instead of optimising for mass it was decided at this stage to optimise for payload deceleration and residual energy in order to achieve the best model to carry forward to the reliability analysis. Note that the same approximation models could be used for the reformulated optimization problem.

38. Optimization results • Optimised mass selected was 403.5kg (baseline 392.8kg) • Flat Bottom Impact Payload Acceleration increased from 65.5g to 67.7g • Rock Impact Payload Acceleration decreased from 980.3g to 69.1g • Maximum Material Stresses were reduced from 281MPa to 157MPa • While 3 variables are in middle of range, Vent Area pushes limit

39. Reliability assessment of ExoMars lander • Ultimately, the reliability figure gives the probability of a successful landing for a given design under a range of conditions • Alternatively it can be used to establish an envelope of conditions for a given success probability • For this project, the limited number of variables (4) considered, results in a reliability index 'figure of merit', rather than an overall probability of success • Establishing this 'figure of merit', index for the reliability of a design gives a useful comparison with alternative designs

40. Reliability assessment: Model definitions • Adopt airbag design variables determined by optimization study • Consider only rock impact loadcase (though rock height may be zero, i.e. flat surface) • Failure defined by exceeding similar constraints to optimisation study • Resultant deceleration < 80g • Kinematic metrics, re-bound, roll-over • No bag tearing • Environment Variables

41. Stochastic Variables & PDFs • Wind Velocity - Weibull distribution, Determined from European Mars Environmental Model Project

42. Stochastic Variables & PDFs • Rock Height -Exponential distribution, Determined from past mission data • Pitch Attitude – Normal distribution - Variation from mean at ±3σ is ±30° • Pitch Rate – Normal distribution - Variation from mean at ±3σ is ±20°/sec

43. Reliability assessment of ExoMars lander: Metamodel generation • Test Plan – Uniform Latin HyperCube with Extremities Extension • Eighty Test Points – 80 LS-DYNA runs executed (single load case) • Advanced metamodelling using moving least squares method • Process is the same as used to generate optimization metamodel, but with different variables

44. Reliability assessment of ExoMars lander Objective: avoid this…

45. Result of reliability assessment of ExoMars lander • Metamodel generation methodology successfully used to perform optimization and assess reliability in Altair’s HyperStudy • The optimization study arrived at a design that satisfies the requirements with only a small increase in mass • Reliability analysis proved that the concept is viable • Reliability analysis uncovered failure modes that had not previously been considered

46. Conclusions • The suggested airbag optimization & reliability assessment process can be employed in future phases of the ExoMars project, with: - Further design improvements by increasing design variable space (vent areas, trigger accelerations, more complex differential venting strategies, changes in the un-vented toroidal, etc.) - More comprehensive reliability assessment, with aim of determining the overall airbag reliability

47. Astrium Mars Lander: LS-DYNA Simulation

48. Questions? Any questions?