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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
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 4 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 5 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 6 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 7 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 8 / 41
Uncertainty Representation
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 10 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2
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60
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90
270
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180 0
Time = 1.00 orbits (1.8 hr)
1
2
30
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60
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90
270
120
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150
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180 0
Time = 5.00 orbits (8.8 hr)
1
2
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270
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150
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180 0
Time = 10.00 orbits (17.5 hr)
1
2
30
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60
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90
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120
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150
330
180 0
Time = 20.00 orbits (35.0 hr)
1
2
30
210
60
240
90
270
120
300
150
330
180 0
Time = 50.00 orbits (87.5 hr)
2016 DDDAS AFOSR Spring Review 11 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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.
2016 DDDAS AFOSR Spring Review 12 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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.
2016 DDDAS AFOSR Spring Review 13 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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.
2016 DDDAS AFOSR Spring Review 14 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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)
2016 DDDAS AFOSR Spring Review 15 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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?
2016 DDDAS AFOSR Spring Review 16 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 17 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 18 / 41
Uncertainty Propagation
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 20 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 21 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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 )
2016 DDDAS AFOSR Spring Review 22 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 23 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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.
2016 DDDAS AFOSR Spring Review 24 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 25 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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).
2016 DDDAS AFOSR Spring Review 26 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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?
2016 DDDAS AFOSR Spring Review 27 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 28 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 29 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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.
2016 DDDAS AFOSR Spring Review 30 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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.
2016 DDDAS AFOSR Spring Review 31 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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.
2016 DDDAS AFOSR Spring Review 32 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
Experimental Validation of P ∗
For this problem P ⋆ = 70
Analytical P ⋆ matches experimental valueThus, synchronization occurs at iterations 70, 140, 210, · · ·
2016 DDDAS AFOSR Spring Review 33 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 34 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
LSDA vs TSDA ConvergenceLSDA converges faster than TSDA
Figure: Dual variable solution (y) vs Number of iterations (k).
2016 DDDAS AFOSR Spring Review 35 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 36 / 41
Summary
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm 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
2016 DDDAS AFOSR Spring Review 38 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
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
2016 DDDAS AFOSR Spring Review 39 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
Summary #3New algorithm for solving large-scale distributed QPs
Relaxed synchronization of Lagrange multipliersRecovers optimal solution160 times speed upImpacts many applications
2016 DDDAS AFOSR Spring Review 40 / 41
Introduction Uncertainty Representation Uncertainty Propagation LSDA algorithm Summary
AcknowledgementsThis research was supported by AFOSR DDDAS grantFA9550-15-1-0071, with Dr. Frederica Darema as the programmanager.
Thank you!
2016 DDDAS AFOSR Spring Review 41 / 41