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Cloud Computing Based Robust Space Situational Awareness

Cloud Computing Based Robust Space Situational Awareness

Spring 2016 Review

Kooktae Lee Aerospace

Niladri Das Aerospace

Riddhi Pratim Ghosh Statistics

Raktim Bhattacharya Aerospace (PI)

Bani Mallick Statistics

Faming Liang Statistics (UFL)

2016 DDDAS AFOSR Spring Review

Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary

OverviewWhat is Space Situational Awareness?

Why is it important? Issues & Challenges What is the impact of uncertainty? DDDAS opportunities

Uncertainty Modeling The right coordinates The right model

Uncertainty Propagation Nonlinear, non Gaussian Real-time predictions Accuracy Computational challenges – scalability & accuracy

2016 DDDAS AFOSR Spring Review 2 / 41

Space SituationalAwareness


What is Space Situational Awareness?Answer for every space object

Where is it?Where is it going?What is it? operational satellite, non operational, rocket body, debris?What is it doing? Status change?

Presented by Raktim Bhattacharya AFOSR Spring Review, Arlington, 2016.

SSA Issues• Space situational awareness

is critical – US SSN can only track 30K of

500K objects• Collision is a real risk– 2009 collision between Iridium-33

and Kosmos-2251– 2013 Pegasus with Soviet debris

• Collision creates debris field– Affects communication– Increases collision risk with other

operational satellites

Image Courtesy NASA



SSA IssuesSpace situational awareness is critical

US Space Surveillance Network (SSN) can only track 30K of500K objects

Collision is a real-risk 2009 collision between Iridium-33 and Kosmos-2251 2013 Pegasus with Soviet debris

Collision creates debris field Affects communication, surveillance and navigation Increases collision risk with other operational satellites

UQ uses and needs Probability of collision Data or track association/correlation Sensor tasking Maneuver detection



SSA ChallengesLarge uncertainty in satellite’s location

Non Gaussian in R5 × SObservations are sparse

Long time gaps =⇒ long propagation times Geopolitical constraints

Spatio-temporal problem Distributed sensing Fast dynamics Spatial nonlinearities

Computationally intensive High accuracy (tails) Real-time prediction

Presented by Raktim Bhattacharya AFOSR Spring Review, Arlington, 2016.

SSA Challenges• Large uncertainty in satellite’s location– Non Gaussian in

• Observations are sparse– Long time gaps– Geopolitical constraints

• Spatio-Temporal problem– Sensing is distributed– Dynamics is fast – Nonlinearities are spatial

• Computationally intensive– High accuracy (tails)– Real-time prediction

R5 S

US Space Surveillance Network



DDDAS OpportunitiesSpace Data Association

Enabling satellite data sharing with USSTRATCOM, foreign,commercial and military stake holders

DARPA’s Orbit Outlook (O2) Program to expand SSN leverage civil, academic, industry and government surveillance

infrastructure SpaceView (amateur astronomers, especially for GEO) StellarView (University infrastructure) Data from LILO (Low Inclined LEO Objects) initiative

Our Research New algorithms to support these initiatives Fuse data from multiple sensors Provide feedback for sensor scheduling



Key Research ObjectivesUncertainty propagation (2016 Spring Review)

representation in cylindrical manifold (R5 × S) scalable nonlinear, non Gaussian methods

Distributed satellite trackingData driven model refinement

Estimation of unknown model parameters (dynamics,perturbation models)

Uncertainty mitigation Collision avoidance High confidence orbit design Resource allocation (sensing scheduling etc.)

Scalable cloud-ready implementation SSA algorithms as a service Cloud-ready =⇒ need based scalability


Uncertainty Representation


The Right CoordinatesDynamics

r = −GME

r3r + apert(r, r, t)

CoordinatesKeplarian orbital elements: (a, e, i,Ω, ω,M)

Numerical instabilities for satellite problemsEquinoctial orbital elements: (a, h, k, p, q, l)

Defines cylindrical coordinate system in R5 × S (Horwood et al. ) (a, h, k, p, q) ∈ R5, defines geometry and orientation of orbit l ∈ S is angular coordinate that defines the location along the

orbitUncertainty

(a, h, k, p, q) are conserved by Kepler’s laws =⇒uncertainties do not growuncertainty in l grows without bound



Uncertainty Growth in Angular Variable lInitial condition uncertainty: Range = 1000± 1.5 km, Angular Position = ±5

1

2

30

210

60

240

90

270

120

300

150

330

180 0

Time = 0.00 orbits (0.0 hr)

1

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180 0


1

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1

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1

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2

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Von Mises DistributionThe Von Mises distribution is defined as

VM(θ,Θ, κ) :=1

2πI0(κ)eκ cos(θ−Θ),

whereΘ is the mean directionκ ≥ 0 is the concentration parameterI0(κ) is modified Bessel function

I0(κ) :=1

2π

∫ 2π

0eκ cos(θ)dθ.



Gauss Von Mises Distribution in R5 × S

Since (a, h, k, p, q, l) ∈ R5 × SHorwood et al. proposed

p(x, θ) = N (x;µ,P )VM(θ,Θ(x), κ),

where x := (a, h, k, p, q)T and θ := l.



Gauss Von Mises DistributionHow well does it capture uncertainty in R5 × S?

Result 1With

p(x, θ) = N (x;µ,P )VM(θ,Θ(x), κ),

marginal distribution of θ over x, i.e.∫xp(x, θ)dx

is not Von Mises.Proof: Compare characteristic functions of

Distribution VM(θ,Θ, κ)∫x p(x, θ)dx

Characteristic Fcn Im(κ)eimµ

I0(κ)Im(κ)eimαe−

12m2βT (I−imΓ)−1β

I0(κ)

Details: Conference paper in preparation.



Gauss Von Mises DistributionHow well does it capture uncertainty in R5 × S?

Result 2But ∫

x∈Rn

p(x, θ)dx

is circular.Proof: Satisfies properties of circular distribution.

1)

∫xp(x, θ)dx ≥ 0,

2)

∫ 2π

0

∫x∈Rn

p(x, θ)dxdθ = 1

3) p(θ) =

∫x∈Rn

p(x, θ)dx =

∫x∈Rn

p(x, θ + 2πk)dx = p(θ + 2πk)

for any integer k, as only periodic term is cos(θ − α)



Few concernsSufficient statistics of ∫

xp(x, θ)dx?

as it is not Von MisesIf uncertainty in θ is VM, how to model uncertainty in (x, θ)as GVM?Does p(t,x, θ) remain GVM?Are there other modeling options?



New Modeling ApproachCoherent Joint Distribution

Model uncertainty in (x,θ) as generalized exponentials

p(x,θ) ∝ exp[−1

2xTΣ−1x+ λTΣ−1x+ a(θ)Σ−1x

]where

x ∈ Rn1 ,

θ ∈ [0, 2π]n2 ,

Σ−1 > 0,

a(θ) := [a1(θ), · · · , an1(θ)]T , with

ai(θ) =

n2∑j=1

n∑k=1

ai cos[k(θj − µijk)], i = 1, · · · , n1.

n is sample size



Properties of Coherent DistributionsArbitrary manifolds x ∈ Rn1 ,θ ∈ Sn2

For SSA, n1 = 5, n2 = 1∫xp(x, θ)dx

is Von Misesp(x,θ) is a member of exponential family with well definedsufficient statisticsUse theory of exponential family for optimal inference &predictionMarginal of x ∫

θp(x, θ)dθ

is n1 dimensional Gaussian with Mean: λ+ a(θ) Covariance: Σ


Uncertainty Propagation


Sources of UncertaintyInitial condition uncertaintyDynamics

r = −GME

r3r + apert(r, r, t)

apert includes Gravitational variation Atmospheric drag Third-body perturbations Solar radiation pressure

System dynamics has both parametric uncertainty and processnoiseSensing is also noisy, especially angle measurements



Uncertainty PropagationDepends on the nature of uncertainty

Dynamicsx = f(t,x,ρ0(1 +∆ρ)︸︷︷︸

ρ

) +w

Only Parametric Uncertainty w = 0

∂p(t,x)

∂t+∇.(pf) = 0 Continuity Equation

Only Process Noise ∆ρ = 0

∂p(t,x)

∂t+∇.(pf)+

1

2∇.Q.(∇p) = 0 Fokker-Planck-Kolmogorov Equation



Parametric Uncertainty PropagationBasic Idea

Dynamicsx = f(t,x,ρ)

PropagationX =

[xρ

],

∂p(t,X)

∂t+∇.(pF ) = 0

Solution Method of Characteristics

X = F (t,X)

p = −p(∇.F )



Parametric Uncertainty PropagationComputational Efficiency

101

102

103

104

10−6

10−5

10−4

10−3

10−2

10−1

100

Convergence of E [x]

Number of samples

Errorin

E[x]

FP−MOCMC

Orders of magnitude more efficient than Monte-CarloTrivially parallelizable – suitable for GPU architectures



Uncertainty Propagation with Process NoiseBasic Idea

Dynamicsx = f(t,x,ρ) + w

PropagationX =

[xρ

],

∂p(t,X)

∂t+∇.(pF ) +

1

2∇.Q.(∇p) = 0

Solution Approximated by mixture of Gaussian Terejanu et al.

Approximate solution p(t,X) ≈∑

i αiN (µi,Σi)

Update µi,Σi using local linear dynamicsUpdate αi by minimizing 2-norm of equation error

miny

yTPy + qTy + s, subject to Ay ≤ b.



Uncertainty Propagation with Process NoiseIssues

QPminy

yTPy + qTy + s, subject to Ay ≤ b.

needs to be solved every ∆t

∆t is small to ensure accurate µi,Σi updates

Problem size can be quite largeNeed real-time predictionsMust be able to solve large QPs fast!

New Research ThrustNew parallel QP solver for multi-core machines



Original QP problem

minx

f(x) :=1

2xTQx+ cTx, subject to Ax = b, (1)

where x ∈ Rn, A ∈ Rm×n, b ∈ Rm, Q ∈ Rn×n is a symmetric,positive definite matrix, and c ∈ Rn.Assume f(x) is separable , i.e.

f(x) :=

N∑i=1

fi(xi), Ax :=

N∑i=1

Aixi,

where N numbers of subproblems (processing elements).



Dual Ascent (DA) MethodDual problem with respect to variable y:

xk+1i = arg min

xi

Li(xi, yk) = −Q−1

i (ATi y

k + c), i = 1, . . . , N

yk+1 = yk + αk(Axk+1 − b),

where Li(xi, y) := fi(xi) + yT (Aixi − bi).

IssuesBefore y-update, x needs to be synchronized

Communication is slower than computationResults in processor idle timeParallelization is not beneficial

How to overcome this?



Dual Ascent (DA) MethodLinear Iterations

xk+1i = −Q−1

i (ATi y

k + c), i = 1, . . . , N

yk+1 = yk + αk(Axk+1 − b),

where Li(xi, y) := fi(xi) + yT (Aixi − bi).

How to overcome communication overhead?Asynchronous Synchronization

solution is stochastichard to prove convergenceharder to verifyexperimentally

Lazy SynchronizationUpdate y at a slower rateReduces communicationoverheadSignificantly faster solutiontime



Lazily Synchronized Dual Ascent AlgorithmUpdate y with period P

Lazy y update

y(t+1)P =

(I − P

N∑i=1

αi

(AiQ

−1i AT

i

))ytP

− PN∑i=1

αi

(AiQ

−1i c+

b

N

), (2)

where P denotes the synchronization period.

Objective: Find optimal P ⋆, such that1) LSDA (2) is numerically stable,2) P ⋆ guarantees the fastest convergence to the solution



Stability ConditionResultThe lazily synchronized dual ascent (LSDA) algorithm is stable ifand only if

ρ

(I − P

N∑i=1

αi

(AiQ

−1i AT

i

))< 1, (3)

where the symbol ρ(·) denotes the spectral radius.Note: Stability condition depends only on partitioned data Ai, Qi.

Proof: See paper.A Relaxed Synchronization Approach for Solving Parallel Quadratic Programming Problems with GuaranteedConvergence, K. Lee, R. Bhattacharya, J. Dass, V. N. S. P. Sakuru, R. Mahapatra, 30th IEEE InternationalParallel & Distributed Processing Symposium.



Optimal Synchronization Period P ⋆

ResultFor the given PQP problem with LSDA technique, the optimalsynchronization period P ⋆ is given by

P ⋆ = max argminP∈N

max|1− λ(β)P |, |1− λ(β)P |, (4)

where λ(·) and λ(·) denote the smallest and the largest eigenvaluesof the square matrix, respectively and β :=

∑Ni=1 αiAiQ

−1i AT

i .

Note: Computation of P ∗ depends only on partitioned data Ai, Qi.

Proof: See paper.A Relaxed Synchronization Approach for Solving Parallel Quadratic Programming Problems with GuaranteedConvergence, K. Lee, R. Bhattacharya, J. Dass, V. N. S. P. Sakuru, R. Mahapatra, 30th IEEE InternationalParallel & Distributed Processing Symposium.



Experimental SetupNumber of processors N = 10, 20, 32, 40,Problem size d = 200, 000

Data generated synthetically as

Qi ∈ RdN× d

N , Ai ∈ R1× dN , c ∈ R

dN×1, xi ∈ R

dN×1 ∀ i = 1, · · · , N.



Experimental Validation of P ∗

For this problem P ⋆ = 70

Analytical P ⋆ matches experimental valueThus, synchronization occurs at iterations 70, 140, 210, · · ·



Solution Time vs Cluster SizeConventional parallelization (TSDA) increases solution time

Figure: Total execution time vs Cluster size.

Communication time dominates computational timeNo benefit from parallelization



LSDA vs TSDA ConvergenceLSDA converges faster than TSDA

Figure: Dual variable solution (y) vs Number of iterations (k).



LSDA vs TSDA Performance SummaryTSDA algorithm LSDA algorithm

Number of iterations 868 211Synchronization period 1 70Number of synchronizations 868 (=868/1) times 3 (=211/70) timesComm. delay reduction 99.65%Speedup 160 times


Summary


Summary #1Modeling uncertainty in the manifold they evolve, reduces computational complexity

Gauss Von-Mises is not coherent, w.r.t marginals

Proposed a new exponential family for arbitrary cylindricalmanifolds Rn1 × Sn2 – Nice properties (coherent, efficientinference & prediction)Can be extended to mixture models



Summary #2Uncertainty Propagation

Parametric uncertainty is simpler to handle Mimics MC, but orders of magnitude more efficient pdf evolves along x(t) Trivially parallelizable (suitable for GPU like architectures)

Process noise introduces complexity (diffusion term) Requires solution of a large QP every ∆t Research focus was to parallelize QP for multicore machines



Summary #3New algorithm for solving large-scale distributed QPs

Relaxed synchronization of Lagrange multipliersRecovers optimal solution160 times speed upImpacts many applications



AcknowledgementsThis research was supported by AFOSR DDDAS grantFA9550-15-1-0071, with Dr. Frederica Darema as the programmanager.

Thank you!


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