A Unified Dynamic Information Guided Particle Framework for Mission ...
A Unified Dynamic Information Guided Particle Framework for Mission Design Mrinal Kumar, The Ohio State University
•
Summary of Effort
– AF Relevance: Enhancing autonomy in multi-agent teams aided by a heterogeneous sensor-field, deployed in an uncertain/ unstructured environment
•
Key Focus of Scientific Research –
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The goal of this research is to create a unified particle platform that enables efficient processing of complex data and communication of extracted information among its constituent algorithms for designing and conducting mission operations such as modeling, sensing, forecasting, assimilation, allocation and control; thereby building an inter-compatible structure consistent with the DDDAS philosophy of unification of the computational and sensing modules in the mission.
Other performers on project – –
Mrinal Kumar, PI Alex Soderlund & Chao Yang (Ph.D. Students), Dr. Donghoon Kim (Post-doctoral associate)
1
A Unified Dynamic Information Guided Particle Framework for Mission Design Mrinal Kumar, The Ohio State University
OVERVIEW of Project Elements
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An Adaptive Particle Paradigm for Uncertainty Forecasting in High-Dimensional Nonlinear Stochastic Systems Mrinal Kumar Associate Professor, MAE Laboratory for Autonomy in Data-Driven and Complex Systems (LADDCS) The Ohio State University [email protected]
September 6, 2017
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
1 / 14
Forecasting Uncertainty: The Fokker-Planck Equation ⊛ Physics (Stochastic Differential Equation): dx = f(t, x)dt + g(t, x)dB(t) [ x ∈ Rp ; B(t) ∈ RM ] ´¹¹ ¹ ¹¸¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ state noise
⊛ Fokker-Planck Equation (FPE): ∂ W(t, x) = LF P [W(t, x)], ∂t W(t0 , x) = W0 (x) ∶ initial condition uncertainty The FP Operator ⎡ P P ⎤ ⎢1 ⎥ ∂ ∂ ∂ ∂2 f1 + f2 + . . . fP ](⋅) + ⎢ (gQgT )⎥ ∑∑ ⎢ ⎥(⋅) ∂x1 ∂x2 ∂xP 2 ∂x ∂x ⎢ i=1 j=1 i j ⎥ ⎣ ⎦ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
LF P (⋅) = − [
Drift
⊛ Primary challenges: ● nonlinearity... ● dimensionality... M. Kumar (L. Autonomy in DD & CS)
Diffusion
... entails non-Gaussianity ... causes scalability issues (w.r.t. P) Unified Particle Paradigm
September 6, 2017
4/5
Forecasting Uncertainty: The Stochastic Liouville Equation ⊛ Physics (ODE model): dx = f(t, x)dt [ x ∈ Rp ] ⊛ Stochastic Liouville Equation (SLE): ∂ W(t, x) = LSL [W(t, x)], ∂t W(t0 , x) = W0 (x) ∶ initial condition uncertainty The SL Operator ∂ ∂ ∂ f1 + f2 + . . . fP ](⋅) ∂x1 ∂x2 ∂xP ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
LSL (⋅) = − [
Drift
⊛ Primary challenges: ● nonlinearity... ● dimensionality...
M. Kumar (L. Autonomy in DD & CS)
... entails non-Gaussianity ... causes scalability issues (w.r.t. P)
Unified Particle Paradigm
September 6, 2017
5/5
Focus Application: Space Situational Awareness Statement “For the purpose of protecting our space assets, provide decision-making support with timely and quantifiable evidence of behaviors attributable to specific space domain threats and hazards. This includes, among many other things, tracking all man made objects in space (∼ resident space objects), as well a natural threats such as potentially hazardous asteroids.”
Evolution of the Space Debris Problem
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
4 / 14
Forecasting with Particles: Forward Monte Carlo ⊛ Dynamics: dx = f(t, x)dt I.C. Uncertainty: x0 ∼ W(t0 , x) = W0 (x) Particle representation: {Xi0 }D i=1 ∼ W0 Followed by evolution through dynamics: D
WtFMC ∼ {Φt (Xi0 )}i=1
Wt = Truth;
Note: WtFMC = FMC approximation of the truth
Question Asked in This Work: Under what circumstances could the propagated FMC ensemble (i.e. WtFMC ) be statistically equivalent to an ensemble drawn by direct sampling of the true, (but unknown!) state pdf, i.e. Wt ? ∗∗
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
This is rigorously true only at t = t0 ..
September 6, 2017
5 / 14
Wt versus WtFMC Question Asked in Previous Slide Under what circumstances could the propagated FMC ensemble (i.e. WtFMC ) be statistically equivalent to an ensemble drawn by direct sampling of the true, unknown, state pdf, i.e. Wt ? To seek answer: Study FMC in terms of evolution of its associated transition kernel.
P0FMC
PtFMC
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
6 / 14
Wt versus WtFMC Question Asked in Previous Slide Under what circumstances could the propagated FMC ensemble (i.e. WtFMC ) be statistically equivalent to an ensemble drawn by direct sampling of the true, unknown, state pdf, i.e. Wt ? To seek answer: Study FMC in terms of evolution of its associated transition kernel.
Theorem 1. Evolution of FMC Kernel: General Case Time propagation of the FMC transition kernel PtFMC through system dynamics (dx = f(t, x)dt) is given by the following integro-differential equation: ⎤ ⎡ ⎢⎛ ∂PtFMC (ε, x) N ∂PtFMC (ε, x) ⎞ FMC ⎥ ⎥ dε = 0 ⎢ + f ⋅ W (ε) ∑ i ∫ ⎢ ⎥ ⎠ t ∂t ∂xi ⎥ ⎢⎝ i=1 ⎦ RN ⎣
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
(1)
6 / 14
Wt versus WtFMC Question Asked in Previous Slide Under what circumstances could the propagated FMC ensemble (i.e. WtFMC ) be statistically equivalent to an ensemble drawn by direct sampling of the true, unknown, state pdf, i.e. Wt ? To seek answer: Study FMC in terms of evolution of its associated transition kernel.
Theorem 1. Evolution of FMC Kernel: General Case Time propagation of the FMC transition kernel PtFMC through system dynamics (dx = f(t, x)dt) is given by the following integro-differential equation: ⎤ ⎡ ⎢⎛ ∂PtFMC (ε, x) N ∂PtFMC (ε, x) ⎞ FMC ⎥ ⎥ dε = 0 ⎢ + f ⋅ W (ε) ∑ i ∫ ⎢ ⎥ ⎠ t ∂t ∂xi ⎥ ⎢⎝ i=1 ⎦ RN ⎣
(1)
Theorem 2. Special Case: ∇ ⋅ f = 0 For systems with zero divergence, a valid FMC transition kernel PtFMC with initial condition P0 , that satisfies Theorem 1. is (2) PtFMC (x, y) = P0 [Φ0 (x), Φ0 (y)].
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
6 / 14
Wt versus WtFMC Question Asked in Previous Slide Under what circumstances could the propagated FMC ensemble (i.e. WtFMC ) be statistically equivalent to an ensemble drawn by direct sampling of the true, unknown, state pdf, i.e. Wt ? To seek answer: Study FMC in terms of evolution of its associated transition kernel.
Theorem 3. Special Case: ∇ ⋅ f = 0 If the system dynamics is divergence-free, i.e. ∇ ⋅ f(t, x) = 0, the propagated FMC transition kernel, PtFMC (x, y), is in detailed balance with WtFMC , which in this case, must equal Wt . 1
In essence, for divergence free systems, the FMC propagated ensemble is statistically equivalent to an ensemble directly sampled from the current, true state pdf.
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No such guarantee is possible for systems with non-zero divergence.
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
6 / 14
Wt versus WtFMC : Results 10 0
FMC-3K MCMC/MOC-3K FMC-5K MCMC/MOC-5K FMC-10K MCMC/MOC-10K
10 -1
log(KL)
log(KL)
10 -1
10 -2
10 -2
10
-3
FMC-3K MCMC/MOC-3K FMC-5K MCMC/MOC-5K FMC-10K MCMC/MOC-10K
10 -4
10 -3
0
1
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3
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10 -5
0
Time (s)
Undamped Duffing Oscillator (Divergence Free System)
10
20
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Time (s)
Reentry Through Earth’s Atmosphere (System with Non-Zero Divergence)
Information Theoretic Distance (Relative Entropy) Between True State pdf and the FMC (solid) and MCMC (dashed) Ensembles. In general, there is need for an Adaptive Framework that a.) measures MC performance; and b.) adds/removes particles depending on current performance M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
7 / 14
Adaptive FMC: Preliminaries (1/2) Key Observations. [for finite sized FMC simulations.] 1
In general, there is no guarantee that the propagated FMC ensemble continues to be statistically equivalent to the true state pdf.
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Even when it does (divergence free systems), its “accuracy” shows transient behavior for a fixed sized ensemble. MEASURING FMC PERFORMANCE ¯ t ) ≐ EW [h(x(t))] Consider h(x t
= can be shown
EW0 [ St (x0 ) ] ´¹¹ ¹ ¹ ¸ ¹ ¹ ¹ ¶ h(Φt (x0 ))
MC approximation: ̃ hD =
1 D
i ∑D i=1 h(Xt ) =
1 D
i ∑D i=1 St (X0 )
√ Then, MC Error (Std. Dev.): σεMC =
¯−̃ E [(h h D )2 ] →
σ(h(xt )) √ D
√ (arbitrarily), set h(xt ) = xt , such that EtD ≐ ¯ − ̃ Another KEY Observation: ∣h h ∣ ≤ D∗ (x0 )V (St )
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
xi ) σ 2 (˜ ∑P i=1 t PD
[Koksma-Hlawka]
September 6, 2017
8 / 14
Adaptive FMC: Preliminaries (2/2) ⊛ Consider an ensemble P = {X1 , X2 , . . . , Xn } in N dimensions, drawn from a uniform distribution on C ≡ ⊗N [0, 1]. ⊛ It is said to be optimally efficient if is has: 1 Space-filling property: Particles that fill the space as homogeneously as possible.. quantified via the Lp discrepancy metric: 1/p ⎧ ⎫ p ⎪ ⎪ ⎪ ⎪ # (Pu , Jxu ) ⎪ ⎪ Dp (P) = ⎨ ∑ ∫ ∣ − Vol (Jxu )∣ dx⎬ ⎪ ⎪ n ⎪ ⎪ ⎪u≠0Cu ⎪ ⎩ ⎭ Generated by Space-filling and Non-Collasping Criteria
X2
1
0.5
0 0
0.2
X2
0.4
0.6
0.8
1
0.8
1
X1 Pseudorandom
1
0.5
0 0
0.2
0.4
0.6 X1
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
9 / 14
Adaptive FMC: Preliminaries (2/2) ⊛ Consider an ensemble P = {X1 , X2 , . . . , Xn } in N dimensions, drawn from a uniform distribution on C ≡ ⊗N [0, 1]. ⊛ It is said to be optimally efficient if is has: 1 Space-filling property: Particles that fill the space as homogeneously as possible.. quantified via the Lp discrepancy metric: 1/p ⎧ ⎫ p ⎪ ⎪ ⎪ ⎪ # (Pu , Jxu ) ⎪ ⎪ Dp (P) = ⎨ ∑ ∫ ∣ − Vol (Jxu )∣ dx⎬ ⎪ ⎪ n ⎪ ⎪ ⎪u≠0Cu ⎪ ⎩ ⎭ 2
Non-collapsing property: Particles do not coincide when projected on to any lower dimensional space
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
9 / 14
Adaptive FMC: Algorithm 1
(arbitrarily) Set error bounds: E L∗ and E U∗ .
2
t Current FMC ensemble: {Xit }D i=1 . (Size: Dt ). (red diamonds.)
3
Estimate current error (approximately! .. via bootstrapping ): EtDt .
4
If EtDt > E U∗ , add particle(s): a.) Begin with a large candidate set (black dots.) b.) Sequentially apply non-collapsing and space-filling criteria: ⎧ ⎪ ⎪Discarded Rank(Pj′ ) = ⎨ ⎪ CL (P ′ )2 ⎪ ⎩ 2 j
M. Kumar (L. Autonomy in DD & CS)
if minxi ∈P ∣∣xi − xcj ∣∣−∞ ≤ dmin (blue squares) if minxi ∈P ∣∣xi − xcj ∣∣−∞ > dmin (pink triangle!)
Unified Particle Paradigm
September 6, 2017
10 / 14
Adaptive FMC: Algorithm 1
(arbitrarily) Set error bounds: E L∗ and E U∗ .
2
t Current FMC ensemble: {Xit }D i=1 . (Size: Dt ). (red diamonds.)
3
Estimate current error (approximately! .. via bootstrapping ): EtDt .
4
If EtDt > E U∗ , add particle(s): a.) Begin with a large candidate set (black dots.) b.) Sequentially apply non-collapsing and space-filling criteria: ⎧ ⎪ ⎪Discarded Rank(Pj′ ) = ⎨ ⎪ CL (P ′ )2 ⎪ ⎩ 2 j
if minxi ∈P ∣∣xi − xcj ∣∣−∞ ≤ dmin (blue squares) if minxi ∈P ∣∣xi − xcj ∣∣−∞ > dmin (pink triangle!)
5
If EtDt < E U∗ , remove particle(s): a.) Compute “relative weight” by solving SLE numerically along particle trajectory b.) Identify particles with least relative significance and eliminate until EtDt > E U∗ .
6
Continue to next time step...
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
10 / 14
Adaptive FMC: Algorithm X⇤0 Select The Particle for Removal According to Their
Accuracy Test: Bootstrapping
W(t, x) Solve SLE to Obtain W(t, x) for each particle
p
Particle Removal Scheme
Y
Identify Regions Satisfying the Non-collapsing Property
Y
Generate New Candidates inside Identified Regions at T0 Rank the New Candidates by Discrepancy and Obtain X⇤0
Particle Addition Scheme for High Dimensional Case
M. Kumar (L. Autonomy in DD & CS)
X⇤0
Forward Propagate to Tt :
ETt < ETL⇤ If ? t N p
ETt > ETUt⇤ If ? N Evolves to Next Time Step i n Tt+1 : {XTt+1 }i=0
Unified Particle Paradigm
Y
Generate a Set of New Candidates Q at T0 If
?
Y Rank The Remaining Candidates by Discrepancy and Obtain X⇤0
Particle Addition Scheme for Low Dimensional Case
September 6, 2017
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Adaptive FMC: Application ⊛ The Lorenz-96 Dynamical System with x ∈ R40 : x˙k = −xk−2 xk−1 + xk−1 xk+1 − xk + F , k = 1, 2, . . . , 40
History of Adaptations to Maintain MC Performance. Comparative Performance of Adaptive and Traditional MC.
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
12 / 14
Adaptive FMC: Application ⊛ The Two-Body Problem (with J2 and Noise Perturbations)
Accuracy in forecasting v1
Accuracy in forecasting x1
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
13 / 14
Summary Uncertainty forecasting is a fundamental problem in stochastic dynamics, for which two main solution paradigms exist: (a.) functional approximation, (b.) particle representation A benchmark ensemble was constructed by combining MCMC and Method of Characteristics. Following initial discretization, propagated particles are statistically equivalent to the true state pdf only in systems with zero divergence. No such guarantee exists for systems with non-zero divergence. Even for systems with zero divergence, MC approximation accuracy is time varying. An adaptive Monte Carlo framework was developed, based on ensemble discrepancy. It is possible to continuously monitor MC performance and adaptively add and remove particles to maintain control of its transient performance. Current efforts are focused on increasing speed of MC accuracy evaluation adaptation to systems with noise integration with sequential state-estimation
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
14 / 14
Rapid Information-Guided Clustering to Enable Sensor Collaboration in Multi-Target Tracking Mrinal Kumar Associate Professor, MAE Laboratory for Autonomy in Data-Driven and Complex Systems (LADDCS) The Ohio State University [email protected]
September 6, 2017
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
1/3
Informa$on-Guided Op$mal Sensor Clustering Objec&ve: Op$mal sensor alloca$on to enable mul$-target tracking within a heterogeneous wireless sensor network
Informa$on-Guided Op$mal Sensor Clustering Objec&ve: Op$mal sensor alloca$on to enable mul$-target tracking within a heterogeneous wireless sensor network Two exis&ng techniques: (i) Euclidean Clustering (aims for efficiency: not op$mal) (ii) Global Node Search (aims to maximize informa$on gain from sensor field: prohibi$ve cost, esp. for mul$-target tracking) Mo&va&ng Challenge:
In mul$-target tracking, balance is needed between informa$on gain and speed of informa$on transfer: Informa&on U&lity Informa&on Cost Tracking Failure can occur on both ends: on the leH: due to poor informa$on quality on the right: due to delays, resul$ng in dropped targets
Informa$on-Guided Op$mal Sensor Clustering Procedure: A sequence of filters that account for:
a) Sensing feasibility b) Computa$onally efficient informa9on u9lity (via Mahalanobis distance) c) Communica9on cost through the known path to base sta$on for each sensor
W ◆ |{z}
WSN
Fk |{z}
Qk |{z}
Feasible Set
Ck |{z}
IG-RC Output
Utility Set
d) Op9mal sensor finally selected in an IDSQ framework: based on minimiza9on of differen9al entropy of the expected posterior Target Mean Target Covariance Range Sensor Bearing Sensor
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x [m]
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y [m]
50
Target Mean Target Covariance Range Sensor Bearing Sensor
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y [m]
y [m]
Target Mean Target Covariance Range Sensor Bearing Sensor
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x [m]
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x [m]
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Informa$on-Guided Op$mal Sensor Clustering Main Result
Theorem: Given a common sensor field, a cluster of n sensors derived from the IG-RC will provide greater or equal informa9on gain than any Euclidean cluster of same size. n X i
P (I R
I iQ
n X i
N 1 X > I E | x = x t , y = yt ) = ↵i N i=1
I iE
↵i =
(
1, 0,
if S ⇤ 2 C \ (C \ E) otherwise
Informa$on-Guided Op$mal Sensor Clustering Mul&-Target Tracking Performance: Three (friendly) Targets
Alg. Run Time [s]
100 sensors per field (3 Targets)
14000
Rapid Euclidean GNS
12000
Alg. Run Time [s]
8000 6000 4000
Alg. Run Time [s]
Trace(J -1 )
10000
2000 0
0
5
10
15
20
25
Time [s]
“Total Volume” of Uncertainty
30
0.08 0.06
RCA CT ( ) ST ( ) RT ( )
0.04 0.02 0
0.03 0.02
1
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ECA CT ( ) ST ( ) RT ( )
0.01 0
0.1
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GNS CT ( ) ST ( ) RT ( )
0.05
0
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Number of Targets
Clustering Cost
Informa$on-Guided Op$mal Sensor Clustering Mul&-Target Tracking Performance 2 Targets
Trace(J -1 )
4000
RCA ECA GNS
3000
1000 0
5
10
15
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30
4 Targets Trace(J -1 )
2
6000 4000 2000 0
Rapid Euclidean GNS
2.5
8000
Trace(J -1 )
400 sensors per field (8 Targets)
2000
0
0
5
10
15
20
25
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1.5
1
6 Targets
10000
Trace(J -1 )
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3
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8000 6000 0
4000
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Time [s] 2000
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Time [s]
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Informa$on-Guided Op$mal Sensor Clustering Summary
• Developed a provably op$mal sensor clustering algorithm in a heterogeneous wireless sensor network • The approach is computa$onally efficient and guarantees beQer informa$on collec$on compared to ECA
• Current research: use sensor data as evidence, in the framework of DempsterShafer theory, in order to determine the level of conflict with other members of a cluster • The evidence will be used to develop measures of sensor health, with the objec&ve of providing guidelines for sensor realignment
PUBLICATIONS Journal 1. Yang, C. and Kumar, M., “On the Effec$veness of Monte Carlo for Ini$al Uncertainty Propaga$on in Dynamical Systems” Automa+ca, accepted 2. Zhao, Z. and Kumar, M., “A Split-Bernstein Approach to Chance-Constrained Op$mal Control”, Journal of Guidance, Control and Dynamics, accepted 3. Soderlund, A. and Kumar, M., “An Informa$on Guided Rapid Clustering Algorithm for Mul$-Target Tracking in a Wireless Sensor Network”, Transac+ons on Aerospace and Electronic Systems, under review 4. Yang, C. and Kumar, M., “Discrepancy Driven Adap$ve Monte Carlo for Uncertainty Forecas$ng in Nonlinear Dynamical Systems”, Probabilis+c Engineering Mechanics, under review
Conference 1. 2. 3. 4. 5.
Soderlund, A., Kumar, M. and Kim, D., “Rapid Clustering for Op$mal Sensor Selec$on in Heterogeneous Wireless Sensor Networks,” Guidance, Naviga+on and Control Conference @SciTech, Orlando, FL, Jan 8-12, 2018 Yang, C. and Kumar, M., “On the Transient Performance of Monte Carlo Simula$ons for Ini$al Uncertainty Forecas$ng,” Conference on Decision and Control, Melbourne, Australia, Dec 12-15, 2017 Soderlund, A. and Kumar, M., “Consensus-based Object Tracking within Heterogeneous Wireless Sensor Networks”, 1st IAA Interna+onal Conference on Space Situa+onal Awareness, Orlando, FL, Nov 13-15 2017 Yang, C. and Kumar, M., “An Adap$ve Monte Carlo Method for Uncertainty Forecas$ng in Perturbed Two-Body Dynamics”, 1st IAA Interna+onal Conference on Space Situa+onal Awareness, Orlando, FL, Nov 13-15 2017 Soderlund, A. and Kumar, M., “Op$miza$on of Target Tracking with a Sensor Network by Using Expected Likelihood Measurements,” InfoSymbio+cs/DDDAS Conference, Harjord, CT, Aug 9-12, 2016
ST
1 IAA
MANUSCRIPTS ARE SOLICITED ON TOPICS RELATED TO SPACE SITUATIONAL AWARENESS INCLUDING BUT NOT LIMITED TO:
CONFERENCE ON
• alternative (non-propulsive) deorbiting technologies • association • cybersecurity in space • debris removal • drag-controlled re-entry • forecasting • identification • information & communication • proximity operations • risk assessment • resource allocation • RSO/NEO sensing • RSO/NEO identification • space weather • space policy • spacecraft control • testing of debris removal systems (e.g. via CubeSats) • tracking
ABSTRACTS DEADLINE June 30th, 2017 Organized by
WWW.ICSSA2017.COM
SPACE SITUATIONAL
AWARENESS DOUBLETREE B Y H I LT O N H O T E L
ORLANDO EAST-UCF AREA ORLANDO, FLORIDA, USA
NOVEMBER 13 TH - 15 TH 2017
KEYNOTE SPEAKERS AT 1st ICSSA
John Horack, OSU
Moriba Jah, UT
Christophe Bonnal, CNES
George Nield, FAA
Hogler Krag, SDO/ESA
•
Summary of Effort
– AF Relevance: Enhancing autonomy in multi-agent teams aided by a heterogeneous sensor-field, deployed in an uncertain/ unstructured environment
•
Key Focus of Scientific Research –
•
The goal of this research is to create a unified particle platform that enables efficient processing of complex data and communication of extracted information among its constituent algorithms for designing and conducting mission operations such as modeling, sensing, forecasting, assimilation, allocation and control; thereby building an inter-compatible structure consistent with the DDDAS philosophy of unification of the computational and sensing modules in the mission.
Other performers on project – –
Mrinal Kumar, PI Alex Soderlund & Chao Yang (Ph.D. Students), Dr. Donghoon Kim (Post-doctoral associate)
1
A Unified Dynamic Information Guided Particle Framework for Mission Design Mrinal Kumar, The Ohio State University
OVERVIEW of Project Elements
2
An Adaptive Particle Paradigm for Uncertainty Forecasting in High-Dimensional Nonlinear Stochastic Systems Mrinal Kumar Associate Professor, MAE Laboratory for Autonomy in Data-Driven and Complex Systems (LADDCS) The Ohio State University [email protected]
September 6, 2017
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
1 / 14
Forecasting Uncertainty: The Fokker-Planck Equation ⊛ Physics (Stochastic Differential Equation): dx = f(t, x)dt + g(t, x)dB(t) [ x ∈ Rp ; B(t) ∈ RM ] ´¹¹ ¹ ¹¸¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ state noise
⊛ Fokker-Planck Equation (FPE): ∂ W(t, x) = LF P [W(t, x)], ∂t W(t0 , x) = W0 (x) ∶ initial condition uncertainty The FP Operator ⎡ P P ⎤ ⎢1 ⎥ ∂ ∂ ∂ ∂2 f1 + f2 + . . . fP ](⋅) + ⎢ (gQgT )⎥ ∑∑ ⎢ ⎥(⋅) ∂x1 ∂x2 ∂xP 2 ∂x ∂x ⎢ i=1 j=1 i j ⎥ ⎣ ⎦ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
LF P (⋅) = − [
Drift
⊛ Primary challenges: ● nonlinearity... ● dimensionality... M. Kumar (L. Autonomy in DD & CS)
Diffusion
... entails non-Gaussianity ... causes scalability issues (w.r.t. P) Unified Particle Paradigm
September 6, 2017
4/5
Forecasting Uncertainty: The Stochastic Liouville Equation ⊛ Physics (ODE model): dx = f(t, x)dt [ x ∈ Rp ] ⊛ Stochastic Liouville Equation (SLE): ∂ W(t, x) = LSL [W(t, x)], ∂t W(t0 , x) = W0 (x) ∶ initial condition uncertainty The SL Operator ∂ ∂ ∂ f1 + f2 + . . . fP ](⋅) ∂x1 ∂x2 ∂xP ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
LSL (⋅) = − [
Drift
⊛ Primary challenges: ● nonlinearity... ● dimensionality...
M. Kumar (L. Autonomy in DD & CS)
... entails non-Gaussianity ... causes scalability issues (w.r.t. P)
Unified Particle Paradigm
September 6, 2017
5/5
Focus Application: Space Situational Awareness Statement “For the purpose of protecting our space assets, provide decision-making support with timely and quantifiable evidence of behaviors attributable to specific space domain threats and hazards. This includes, among many other things, tracking all man made objects in space (∼ resident space objects), as well a natural threats such as potentially hazardous asteroids.”
Evolution of the Space Debris Problem
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
4 / 14
Forecasting with Particles: Forward Monte Carlo ⊛ Dynamics: dx = f(t, x)dt I.C. Uncertainty: x0 ∼ W(t0 , x) = W0 (x) Particle representation: {Xi0 }D i=1 ∼ W0 Followed by evolution through dynamics: D
WtFMC ∼ {Φt (Xi0 )}i=1
Wt = Truth;
Note: WtFMC = FMC approximation of the truth
Question Asked in This Work: Under what circumstances could the propagated FMC ensemble (i.e. WtFMC ) be statistically equivalent to an ensemble drawn by direct sampling of the true, (but unknown!) state pdf, i.e. Wt ? ∗∗
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
This is rigorously true only at t = t0 ..
September 6, 2017
5 / 14
Wt versus WtFMC Question Asked in Previous Slide Under what circumstances could the propagated FMC ensemble (i.e. WtFMC ) be statistically equivalent to an ensemble drawn by direct sampling of the true, unknown, state pdf, i.e. Wt ? To seek answer: Study FMC in terms of evolution of its associated transition kernel.
P0FMC
PtFMC
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
6 / 14
Wt versus WtFMC Question Asked in Previous Slide Under what circumstances could the propagated FMC ensemble (i.e. WtFMC ) be statistically equivalent to an ensemble drawn by direct sampling of the true, unknown, state pdf, i.e. Wt ? To seek answer: Study FMC in terms of evolution of its associated transition kernel.
Theorem 1. Evolution of FMC Kernel: General Case Time propagation of the FMC transition kernel PtFMC through system dynamics (dx = f(t, x)dt) is given by the following integro-differential equation: ⎤ ⎡ ⎢⎛ ∂PtFMC (ε, x) N ∂PtFMC (ε, x) ⎞ FMC ⎥ ⎥ dε = 0 ⎢ + f ⋅ W (ε) ∑ i ∫ ⎢ ⎥ ⎠ t ∂t ∂xi ⎥ ⎢⎝ i=1 ⎦ RN ⎣
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
(1)
6 / 14
Wt versus WtFMC Question Asked in Previous Slide Under what circumstances could the propagated FMC ensemble (i.e. WtFMC ) be statistically equivalent to an ensemble drawn by direct sampling of the true, unknown, state pdf, i.e. Wt ? To seek answer: Study FMC in terms of evolution of its associated transition kernel.
Theorem 1. Evolution of FMC Kernel: General Case Time propagation of the FMC transition kernel PtFMC through system dynamics (dx = f(t, x)dt) is given by the following integro-differential equation: ⎤ ⎡ ⎢⎛ ∂PtFMC (ε, x) N ∂PtFMC (ε, x) ⎞ FMC ⎥ ⎥ dε = 0 ⎢ + f ⋅ W (ε) ∑ i ∫ ⎢ ⎥ ⎠ t ∂t ∂xi ⎥ ⎢⎝ i=1 ⎦ RN ⎣
(1)
Theorem 2. Special Case: ∇ ⋅ f = 0 For systems with zero divergence, a valid FMC transition kernel PtFMC with initial condition P0 , that satisfies Theorem 1. is (2) PtFMC (x, y) = P0 [Φ0 (x), Φ0 (y)].
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
6 / 14
Wt versus WtFMC Question Asked in Previous Slide Under what circumstances could the propagated FMC ensemble (i.e. WtFMC ) be statistically equivalent to an ensemble drawn by direct sampling of the true, unknown, state pdf, i.e. Wt ? To seek answer: Study FMC in terms of evolution of its associated transition kernel.
Theorem 3. Special Case: ∇ ⋅ f = 0 If the system dynamics is divergence-free, i.e. ∇ ⋅ f(t, x) = 0, the propagated FMC transition kernel, PtFMC (x, y), is in detailed balance with WtFMC , which in this case, must equal Wt . 1
In essence, for divergence free systems, the FMC propagated ensemble is statistically equivalent to an ensemble directly sampled from the current, true state pdf.
2
No such guarantee is possible for systems with non-zero divergence.
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
6 / 14
Wt versus WtFMC : Results 10 0
FMC-3K MCMC/MOC-3K FMC-5K MCMC/MOC-5K FMC-10K MCMC/MOC-10K
10 -1
log(KL)
log(KL)
10 -1
10 -2
10 -2
10
-3
FMC-3K MCMC/MOC-3K FMC-5K MCMC/MOC-5K FMC-10K MCMC/MOC-10K
10 -4
10 -3
0
1
2
3
4
5
10 -5
0
Time (s)
Undamped Duffing Oscillator (Divergence Free System)
10
20
30
40
50
60
70
Time (s)
Reentry Through Earth’s Atmosphere (System with Non-Zero Divergence)
Information Theoretic Distance (Relative Entropy) Between True State pdf and the FMC (solid) and MCMC (dashed) Ensembles. In general, there is need for an Adaptive Framework that a.) measures MC performance; and b.) adds/removes particles depending on current performance M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
7 / 14
Adaptive FMC: Preliminaries (1/2) Key Observations. [for finite sized FMC simulations.] 1
In general, there is no guarantee that the propagated FMC ensemble continues to be statistically equivalent to the true state pdf.
2
Even when it does (divergence free systems), its “accuracy” shows transient behavior for a fixed sized ensemble. MEASURING FMC PERFORMANCE ¯ t ) ≐ EW [h(x(t))] Consider h(x t
= can be shown
EW0 [ St (x0 ) ] ´¹¹ ¹ ¹ ¸ ¹ ¹ ¹ ¶ h(Φt (x0 ))
MC approximation: ̃ hD =
1 D
i ∑D i=1 h(Xt ) =
1 D
i ∑D i=1 St (X0 )
√ Then, MC Error (Std. Dev.): σεMC =
¯−̃ E [(h h D )2 ] →
σ(h(xt )) √ D
√ (arbitrarily), set h(xt ) = xt , such that EtD ≐ ¯ − ̃ Another KEY Observation: ∣h h ∣ ≤ D∗ (x0 )V (St )
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
xi ) σ 2 (˜ ∑P i=1 t PD
[Koksma-Hlawka]
September 6, 2017
8 / 14
Adaptive FMC: Preliminaries (2/2) ⊛ Consider an ensemble P = {X1 , X2 , . . . , Xn } in N dimensions, drawn from a uniform distribution on C ≡ ⊗N [0, 1]. ⊛ It is said to be optimally efficient if is has: 1 Space-filling property: Particles that fill the space as homogeneously as possible.. quantified via the Lp discrepancy metric: 1/p ⎧ ⎫ p ⎪ ⎪ ⎪ ⎪ # (Pu , Jxu ) ⎪ ⎪ Dp (P) = ⎨ ∑ ∫ ∣ − Vol (Jxu )∣ dx⎬ ⎪ ⎪ n ⎪ ⎪ ⎪u≠0Cu ⎪ ⎩ ⎭ Generated by Space-filling and Non-Collasping Criteria
X2
1
0.5
0 0
0.2
X2
0.4
0.6
0.8
1
0.8
1
X1 Pseudorandom
1
0.5
0 0
0.2
0.4
0.6 X1
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
9 / 14
Adaptive FMC: Preliminaries (2/2) ⊛ Consider an ensemble P = {X1 , X2 , . . . , Xn } in N dimensions, drawn from a uniform distribution on C ≡ ⊗N [0, 1]. ⊛ It is said to be optimally efficient if is has: 1 Space-filling property: Particles that fill the space as homogeneously as possible.. quantified via the Lp discrepancy metric: 1/p ⎧ ⎫ p ⎪ ⎪ ⎪ ⎪ # (Pu , Jxu ) ⎪ ⎪ Dp (P) = ⎨ ∑ ∫ ∣ − Vol (Jxu )∣ dx⎬ ⎪ ⎪ n ⎪ ⎪ ⎪u≠0Cu ⎪ ⎩ ⎭ 2
Non-collapsing property: Particles do not coincide when projected on to any lower dimensional space
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
9 / 14
Adaptive FMC: Algorithm 1
(arbitrarily) Set error bounds: E L∗ and E U∗ .
2
t Current FMC ensemble: {Xit }D i=1 . (Size: Dt ). (red diamonds.)
3
Estimate current error (approximately! .. via bootstrapping ): EtDt .
4
If EtDt > E U∗ , add particle(s): a.) Begin with a large candidate set (black dots.) b.) Sequentially apply non-collapsing and space-filling criteria: ⎧ ⎪ ⎪Discarded Rank(Pj′ ) = ⎨ ⎪ CL (P ′ )2 ⎪ ⎩ 2 j
M. Kumar (L. Autonomy in DD & CS)
if minxi ∈P ∣∣xi − xcj ∣∣−∞ ≤ dmin (blue squares) if minxi ∈P ∣∣xi − xcj ∣∣−∞ > dmin (pink triangle!)
Unified Particle Paradigm
September 6, 2017
10 / 14
Adaptive FMC: Algorithm 1
(arbitrarily) Set error bounds: E L∗ and E U∗ .
2
t Current FMC ensemble: {Xit }D i=1 . (Size: Dt ). (red diamonds.)
3
Estimate current error (approximately! .. via bootstrapping ): EtDt .
4
If EtDt > E U∗ , add particle(s): a.) Begin with a large candidate set (black dots.) b.) Sequentially apply non-collapsing and space-filling criteria: ⎧ ⎪ ⎪Discarded Rank(Pj′ ) = ⎨ ⎪ CL (P ′ )2 ⎪ ⎩ 2 j
if minxi ∈P ∣∣xi − xcj ∣∣−∞ ≤ dmin (blue squares) if minxi ∈P ∣∣xi − xcj ∣∣−∞ > dmin (pink triangle!)
5
If EtDt < E U∗ , remove particle(s): a.) Compute “relative weight” by solving SLE numerically along particle trajectory b.) Identify particles with least relative significance and eliminate until EtDt > E U∗ .
6
Continue to next time step...
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
10 / 14
Adaptive FMC: Algorithm X⇤0 Select The Particle for Removal According to Their
Accuracy Test: Bootstrapping
W(t, x) Solve SLE to Obtain W(t, x) for each particle
p
Particle Removal Scheme
Y
Identify Regions Satisfying the Non-collapsing Property
Y
Generate New Candidates inside Identified Regions at T0 Rank the New Candidates by Discrepancy and Obtain X⇤0
Particle Addition Scheme for High Dimensional Case
M. Kumar (L. Autonomy in DD & CS)
X⇤0
Forward Propagate to Tt :
ETt < ETL⇤ If ? t N p
ETt > ETUt⇤ If ? N Evolves to Next Time Step i n Tt+1 : {XTt+1 }i=0
Unified Particle Paradigm
Y
Generate a Set of New Candidates Q at T0 If
?
Y Rank The Remaining Candidates by Discrepancy and Obtain X⇤0
Particle Addition Scheme for Low Dimensional Case
September 6, 2017
11 / 14
Adaptive FMC: Application ⊛ The Lorenz-96 Dynamical System with x ∈ R40 : x˙k = −xk−2 xk−1 + xk−1 xk+1 − xk + F , k = 1, 2, . . . , 40
History of Adaptations to Maintain MC Performance. Comparative Performance of Adaptive and Traditional MC.
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
12 / 14
Adaptive FMC: Application ⊛ The Two-Body Problem (with J2 and Noise Perturbations)
Accuracy in forecasting v1
Accuracy in forecasting x1
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
13 / 14
Summary Uncertainty forecasting is a fundamental problem in stochastic dynamics, for which two main solution paradigms exist: (a.) functional approximation, (b.) particle representation A benchmark ensemble was constructed by combining MCMC and Method of Characteristics. Following initial discretization, propagated particles are statistically equivalent to the true state pdf only in systems with zero divergence. No such guarantee exists for systems with non-zero divergence. Even for systems with zero divergence, MC approximation accuracy is time varying. An adaptive Monte Carlo framework was developed, based on ensemble discrepancy. It is possible to continuously monitor MC performance and adaptively add and remove particles to maintain control of its transient performance. Current efforts are focused on increasing speed of MC accuracy evaluation adaptation to systems with noise integration with sequential state-estimation
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
14 / 14
Rapid Information-Guided Clustering to Enable Sensor Collaboration in Multi-Target Tracking Mrinal Kumar Associate Professor, MAE Laboratory for Autonomy in Data-Driven and Complex Systems (LADDCS) The Ohio State University [email protected]
September 6, 2017
M. Kumar (L. Autonomy in DD & CS)
Unified Particle Paradigm
September 6, 2017
1/3
Informa$on-Guided Op$mal Sensor Clustering Objec&ve: Op$mal sensor alloca$on to enable mul$-target tracking within a heterogeneous wireless sensor network
Informa$on-Guided Op$mal Sensor Clustering Objec&ve: Op$mal sensor alloca$on to enable mul$-target tracking within a heterogeneous wireless sensor network Two exis&ng techniques: (i) Euclidean Clustering (aims for efficiency: not op$mal) (ii) Global Node Search (aims to maximize informa$on gain from sensor field: prohibi$ve cost, esp. for mul$-target tracking) Mo&va&ng Challenge:
In mul$-target tracking, balance is needed between informa$on gain and speed of informa$on transfer: Informa&on U&lity Informa&on Cost Tracking Failure can occur on both ends: on the leH: due to poor informa$on quality on the right: due to delays, resul$ng in dropped targets
Informa$on-Guided Op$mal Sensor Clustering Procedure: A sequence of filters that account for:
a) Sensing feasibility b) Computa$onally efficient informa9on u9lity (via Mahalanobis distance) c) Communica9on cost through the known path to base sta$on for each sensor
W ◆ |{z}
WSN
Fk |{z}
Qk |{z}
Feasible Set
Ck |{z}
IG-RC Output
Utility Set
d) Op9mal sensor finally selected in an IDSQ framework: based on minimiza9on of differen9al entropy of the expected posterior Target Mean Target Covariance Range Sensor Bearing Sensor
70
60
60
50
40
40
30
30
30
40
50
x [m]
60
70
y [m]
50
Target Mean Target Covariance Range Sensor Bearing Sensor
70
60
y [m]
y [m]
Target Mean Target Covariance Range Sensor Bearing Sensor
70
50
40
30
30
40
50
x [m]
60
70
30
40
50
x [m]
60
70
Informa$on-Guided Op$mal Sensor Clustering Main Result
Theorem: Given a common sensor field, a cluster of n sensors derived from the IG-RC will provide greater or equal informa9on gain than any Euclidean cluster of same size. n X i
P (I R
I iQ
n X i
N 1 X > I E | x = x t , y = yt ) = ↵i N i=1
I iE
↵i =
(
1, 0,
if S ⇤ 2 C \ (C \ E) otherwise
Informa$on-Guided Op$mal Sensor Clustering Mul&-Target Tracking Performance: Three (friendly) Targets
Alg. Run Time [s]
100 sensors per field (3 Targets)
14000
Rapid Euclidean GNS
12000
Alg. Run Time [s]
8000 6000 4000
Alg. Run Time [s]
Trace(J -1 )
10000
2000 0
0
5
10
15
20
25
Time [s]
“Total Volume” of Uncertainty
30
0.08 0.06
RCA CT ( ) ST ( ) RT ( )
0.04 0.02 0
0.03 0.02
1
2
3
4
5
6
4
5
6
4
5
6
ECA CT ( ) ST ( ) RT ( )
0.01 0
0.1
1
2
3
GNS CT ( ) ST ( ) RT ( )
0.05
0
1
2
3
Number of Targets
Clustering Cost
Informa$on-Guided Op$mal Sensor Clustering Mul&-Target Tracking Performance 2 Targets
Trace(J -1 )
4000
RCA ECA GNS
3000
1000 0
5
10
15
20
25
30
4 Targets Trace(J -1 )
2
6000 4000 2000 0
Rapid Euclidean GNS
2.5
8000
Trace(J -1 )
400 sensors per field (8 Targets)
2000
0
0
5
10
15
20
25
30
1.5
1
6 Targets
10000
Trace(J -1 )
104
3
0.5
8000 6000 0
4000
0
50
100
150
Time [s] 2000
0
5
10
15
Time [s]
20
25
30
200
250
300
Informa$on-Guided Op$mal Sensor Clustering Summary
• Developed a provably op$mal sensor clustering algorithm in a heterogeneous wireless sensor network • The approach is computa$onally efficient and guarantees beQer informa$on collec$on compared to ECA
• Current research: use sensor data as evidence, in the framework of DempsterShafer theory, in order to determine the level of conflict with other members of a cluster • The evidence will be used to develop measures of sensor health, with the objec&ve of providing guidelines for sensor realignment
PUBLICATIONS Journal 1. Yang, C. and Kumar, M., “On the Effec$veness of Monte Carlo for Ini$al Uncertainty Propaga$on in Dynamical Systems” Automa+ca, accepted 2. Zhao, Z. and Kumar, M., “A Split-Bernstein Approach to Chance-Constrained Op$mal Control”, Journal of Guidance, Control and Dynamics, accepted 3. Soderlund, A. and Kumar, M., “An Informa$on Guided Rapid Clustering Algorithm for Mul$-Target Tracking in a Wireless Sensor Network”, Transac+ons on Aerospace and Electronic Systems, under review 4. Yang, C. and Kumar, M., “Discrepancy Driven Adap$ve Monte Carlo for Uncertainty Forecas$ng in Nonlinear Dynamical Systems”, Probabilis+c Engineering Mechanics, under review
Conference 1. 2. 3. 4. 5.
Soderlund, A., Kumar, M. and Kim, D., “Rapid Clustering for Op$mal Sensor Selec$on in Heterogeneous Wireless Sensor Networks,” Guidance, Naviga+on and Control Conference @SciTech, Orlando, FL, Jan 8-12, 2018 Yang, C. and Kumar, M., “On the Transient Performance of Monte Carlo Simula$ons for Ini$al Uncertainty Forecas$ng,” Conference on Decision and Control, Melbourne, Australia, Dec 12-15, 2017 Soderlund, A. and Kumar, M., “Consensus-based Object Tracking within Heterogeneous Wireless Sensor Networks”, 1st IAA Interna+onal Conference on Space Situa+onal Awareness, Orlando, FL, Nov 13-15 2017 Yang, C. and Kumar, M., “An Adap$ve Monte Carlo Method for Uncertainty Forecas$ng in Perturbed Two-Body Dynamics”, 1st IAA Interna+onal Conference on Space Situa+onal Awareness, Orlando, FL, Nov 13-15 2017 Soderlund, A. and Kumar, M., “Op$miza$on of Target Tracking with a Sensor Network by Using Expected Likelihood Measurements,” InfoSymbio+cs/DDDAS Conference, Harjord, CT, Aug 9-12, 2016
ST
1 IAA
MANUSCRIPTS ARE SOLICITED ON TOPICS RELATED TO SPACE SITUATIONAL AWARENESS INCLUDING BUT NOT LIMITED TO:
CONFERENCE ON
• alternative (non-propulsive) deorbiting technologies • association • cybersecurity in space • debris removal • drag-controlled re-entry • forecasting • identification • information & communication • proximity operations • risk assessment • resource allocation • RSO/NEO sensing • RSO/NEO identification • space weather • space policy • spacecraft control • testing of debris removal systems (e.g. via CubeSats) • tracking
ABSTRACTS DEADLINE June 30th, 2017 Organized by
WWW.ICSSA2017.COM
SPACE SITUATIONAL
AWARENESS DOUBLETREE B Y H I LT O N H O T E L
ORLANDO EAST-UCF AREA ORLANDO, FLORIDA, USA
NOVEMBER 13 TH - 15 TH 2017
KEYNOTE SPEAKERS AT 1st ICSSA
John Horack, OSU
Moriba Jah, UT
Christophe Bonnal, CNES
George Nield, FAA
Hogler Krag, SDO/ESA
