Design optimization and probabilistic assessment of a vented airbag

landing system for the ExoMarsspace mission

Vassili ToropovProfessor of Aerospace and Structural Engineering

Principal Design Optimization Specialist, Altair Engineering (until 2006)

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 by Ariane 5 or Soyuz in 2013

Currently in Phase B – mission planning and concept design phase

ExoMars Space Mission

Airbags for Space Landers Un-vented type

(inflatable ball)• Multiple bounces• Established

heritage (from Luna-9 in 1966)

• High mass• Vulnerable to

rupture

Mars Pathfinder NASA/JPL Beagle 2 Beagle2

Luna 9 USSR Space Program

Airbags for Space Landers

Vented type• Active control• Single stroke• No space heritage• Low mass• Vulnerable to

over-turning

ExoMars ESA

Kistler Booster Irvin

Design concept considers vented (or “Dead-Beat”) airbag coming to rest on second bounce

Inflated with N2 during descent under main parachute

Stowed rover mounted to platform

Vent patches activated by pyrotechnic cutters

Simple reactive vent control system: simultaneous all-vent trigger at 65g

Airbag Landing Design Concept

Six identical vented chambers

One “anti-bottoming” un-vented toroidal

Airbag Configuration

Study Objectives

Develop methodology for optimisation and probabilistic reliability assessment of vented airbags

Key questions for ExoMars:

1. What is the mass of an optimized vented airbag?

2. What is the probability of a successful landing?

3. What is the sensitivity of landing reliability to changing landing scenarios?

Failure modes:• Roll-over (payload overturns),• Dive-through (payload impacts rock)• Rupture (fabric tears)

Full-scale terrestrial testing is difficult / expensive

Optimization and reliability assessment of ExoMars lander

Analysis Overview

DoE

Surrogate Response Surfaces

Optimisation

Design Variables

DoE

Surrogate Response Surfaces

Monte Carlo Simulations

External Variables

ReliabilityOptimum Design

OPTIMISATION ROBUSTNESS

FE Analysis FE Analysis

Explicit FE analysis of 100ms + impact 10 CPU hours +

Metamodelling approach Metamodels built by MLSM Same approach for

optimization and reliability analyses

Optimization variables• Airbag size, pressure, vent

areas Reliability variables

• Wind speed, rock size, pitch attitude & rate

Finite Element Modelling

HyperMesh / LS-DYNA Rigid rover & platform Silicone-coated Vectran airbag

fabric N2 gas

Simulation Stages1. Inflation 2. Initial Impact (65g vent trigger

filtering)3. Venting

Vent Triggering

65g acceleration threshold Measured at payload CoM Signal noise premature venting Crude filter = cumulative time above threshold

Trigger Occurs at 0.0304s

0.0302s65.3g

0.0303s65.7g

0.0304s65.7g

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 building 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

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

Landing Scenarios• Flat Bottom Landing

• Inclined Rock Impact

Mars Environment and Landing Scenarios

Flat Bottom Landing• Vertical velocity 25m/s• Favours ‘tall’ airbag designs• Favours ‘narrow’ airbag designs• Tall, narrow airbag makes most

effective vertical energy absorber

Landing scenarios are chosen to give conflicting design requirements

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

Landing Scenarios

• Peak filtered deceleration 66g (Target <70g)• Peak airbag material stresses 135MPa (Target <533MPa)• Constraints satisfied by baseline design

Baseline Response: Flat Bottom Landing

• Peak filtered deceleration 980g (Target <70g)• Peak airbag material stresses 281MPa (Target <533MPa)• Deceleration constraint exceeded due to ‘Dive Through’

Baseline Response: Inclined Rock Impact

• It is 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’

Dive ThroughDive Through

ExoMars Lander: LS-DYNA Simulation

Design Objective - Minimise system mass (Airbag + Payload + Gas + System)

Design Constraints - Payload acceleration (<70g)- Airbag von Mises stress (<533MPa)- Re-bound and roll over inversion kinematics

Design objective and constraints evaluated for each landing scenario

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

Optimization Set-up

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

¼ ellipse

¼ ellipse

¼ ellipse

Design Parameterization

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

Metamodelling

DOEs for Metamodel Building

Main requirements to a Design of Experiments (DOE) are:• maximum quantity of information• achieved with minimum computational effort (number of numerical

experiments)

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.

LH DOE – conventional (random) and optimal

Random Latin hypercube

Optimal Latin hypercube

DOE Optimization: Objective Function (Audze & Eglais 1977)

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:

= = +

∑ ∑ 21 1

1P P

p q p pqLUmin = min

Potential energy (objective function) Distance between points p and q

permGA Iteration History for 2 Design Variables & 50 Points

permGA Iteration History for 2 Design Variables & 400 Points

Extended Uniform Latin Hypercube

Extended Uniform Latin Hypercube sampling, 21 point + 3 existing points

Extended Uniform Latin Hypercube sampling, 21 point + corners

Uniform Latin Hypercube sampling, 21 point

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New points Existing Points

Four Design Variables – 40 DOE Points (EULH) per Landing Scenario (80 in total)

Metamodelling: DoE

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

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.

xi

xj

rk

DoE point

Evaluation point x

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.

( ) ( ) ( )[ ] min~)()(1

2→−= ∑

=

P

pppp FFwG a,xxxxa

Gaussian weight decay function

wi = exp(-θri2)

where θ is “closeness of fit parameter”

θ = 1 θ = 10 θ = 100

Example: six-hump camel back functionF(x1,x2) = (4 - 2.1 x1

2 + x14 / 3) x1

2+x1x2+(- 4 + 4x22) x2

2, -2 ≤ x1 ≤ 2, -1 ≤ x2 ≤ 1.

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-1

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Example: six-hump camel back function

Original function Approximation on 20 sampling points, θ = 10

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Example: six-hump camel back function

Original function Approximation on 100 sampling points, θ = 120

-2

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Example: Computing and Rendering Point Set Surfaces by M. Alexa et al. 2001

larger θ smaller θ

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 largeramount of noise

Direct Payload to Rock/Ground Impact resulting in high payload accelerations (>100g) occurred at high percentage of DOE points

These high results ‘swamp’ the responses of interest in the approximation

Suggestion: cap response at 100g

Metamodel Generation using MLSM

Noise outside area of interest

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40-4535-4030-3525-3020-2515-2010-155-100-5

To minimise this function, a good quality approximation of the valley should be obtained whilst ignoring numerical noise outside the valley

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Function capped

MLSM Illustrative Example: Rosenbrock’s Banana Valley Function

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Least Squares approximation of capped function, 100 sampling points, quadratic polynomial (left) and cubic polynomial (right), still give poor approximation of function

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MLSM Illustrative Example: Rosenbrock’s Banana Valley Function

Capped function0

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MLSM approximation of capped function, 100 sampling points gives good approximation of function

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MLSM Illustrative Example: Rosenbrock’s Banana Valley Function

Optimization results

• Optimized Mass 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 upper limit

• 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 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)

Optimization results (cont.)

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

Reliability Assessment of ExoMars Lander

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

Design Variable Description

Lower Bound Upper Bound

DV1 Lateral Wind Velocity 0 m/s 20 m/sDV2 Rock Height 0 m 0.8 mDV3 Lander Pitch Attitude -20 º 20 ºDV4 Lander Pitch Rate -30 º/s 30 º/s

Reliability Assessment: Model Definitions

Wind Speed Probability Distribution

European Mars Climate Database (EMCD) - general circulation model

45°N to 45°S latitudes Season 12 Mars Global Surveyor dust

loading scenario PDF fit to EMCD model data Weibull distribution Mean = 8.0 m/s SD = 3.7 m/s

Mean Resultant Wind Speed

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Rock Height Probability Distribution

NASA/JPL rock size distribution model

Viking 1 & 2, MPF landing sites + Earth analogues

Landing Site rock coverage ≤20%

Overall rock coverage from orbital thermal imaging

Rock height = 0.5 x diameter Exponential PDF Mean = 0.196 m SD = 0.196 m

Probability Density Function f(H)

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Rock Height H (m)

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^-1)

k = 10%k = 20%k = 30%

Mars Pathfinder landing site panorama NASA/JPL

Pitch Angle and Pitch Rate

Pendulum motion + gust reaction under parachute at landing Assumed to be random with independent Gaussian Normal PDFs

Pitch Angle Mean = 0 degs 3σ = 30 degs

Pitch Rate Mean = 0 deg/s 3σ = 20 deg/s

• DOE – Uniform Latin HyperCube with Extremities Extension

• Eighty Test Points – 80 LS-DYNA runs executed (single load case)

• Metamodelling using Moving Least Squares Method

• Process is the same as used to generate optimization metamodel, but with different variables

Reliability Assessment of ExoMars Lander: Metamodel generation

Another one bites the dust!

Counting failures…..

Baseline Reliability Analysis & Results

Sample Size Success Probability 100 67.0% 1000 68.3% 10000 68.6%

100, 1000 and 10000 point Monte Carlo simulations

5 mins cpu on PC for 10000 sample points

Low reliability ~ 69% Inversion failure mode is

primary contributor Failed Monte Carlo samples +

FE runs indicate trailing edge contact problem

Sensitivity to Relaxed Landing Conditions

PDF Changed Success Probability Baseline 68.6% Wind Velocity 70.7% Rock height 70.8% Pitch Angle & Rate 80.8%

Reduced latitude band 45°N to 10°S

Morning landing for lighter winds

Landing sites with ≤ 10% rock coverage

Pitch angle and pitch rate variability reduced (knowledge / ‘chute design)

Cumulative improvement in reliability to ~ 81%

Pitch Angle and Rate yield biggest improvement

Parameter PDF Wind Velocity VH Weibull

α = 2 β = 7.49 m/s

Rock Height h Exponential

β = 0.151m

Pitch Angle θ0 Gaussian

3σ = 20°

Pitch Rate Ω0 Gaussian

3σ = 15°/s

Conclusions

General Successful development of methodology to vented

airbags Valuable tool for

• Design Optimization• Probabilistic robustness / reliability analysis• Can be extended to probabilistic design optimization

Importance of simulation-based methods because of difficulty / high cost of representative testing on earth

ConclusionsExoMars application The optimization study arrived at a design that satisfies the

requirements with only a small increase in mass Poor reliability estimated: 69 – 81% Vent areas could be increased Change in un-vented toroidal More complex venting control (e.g. differential triggering)

likely Reliability analysis uncovered failure modes that had not

previously been considered Consider pitch-up and pitch-down in future optimisation Increase simulation times to reduce inversion criteria

uncertainties Reliability analysis process can be employed in future phases

of the ExoMars project, with more comprehensive reliability analyses aiming at determining the overall airbag reliability

Astrium Mars Lander: LS-DYNA Simulation

Any questions?