Scheduling of Multistatic Sonobuoy Fields using Multi

Sonobuoy Fields Tracking Framework Multi-Objective Scheduling Results

Scheduling of Multistatic Sonobuoy Fields using Multi-Objective Optimization Christopher Gilliam1 , Daniel Angley2 , Sofia Suvorova2 , Branko Ristic1 , Bill Moran2 , Fiona Fletcher3 , Han Gaetjens3 , Sergey Simakov3 2

1 School of Engineering, RMIT University, Australia Electrical & Electronic Engineering, The University of Melbourne, Australia 3 Maritime Division, Defence Science and Technology Group, Australia

17th April 2018 Gilliam et al.

Scheduling Multistatic Sonobuoy Fields

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Outline 1

Multistatic Sonobuoy Fields Two Tasks =⇒ Search for and track underwater targets Performance dependent on scheduling sonobuoys

2

Recap on Tracking in Sonobuoy Fields Geometric Modelling and Measurements Tracking algorithm used to track targets

3

Multi-Objective Scheduling Framework Optimization Problem =⇒ Two reward functions Tracking Reward Function Search Reward Function

4

Simulation Results

5

Conclusions

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Track & Search Tasks Scheduling

Multistatic Sonobuoy Fields A network of transmitters and sensors distributed across a large search region

Sensor

Transmitter

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Multistatic Sonobuoy Fields Two tasks of the system: Detect targets that are unknown to the system Sensor

Transmitter

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Multistatic Sonobuoy Fields Two tasks of the system: Detect targets that are unknown to the system Sensor

Transmitter

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Multistatic Sonobuoy Fields Two tasks of the system: Detect targets that are unknown to the system Accurately track targets known to the system Sensor

Transmitter

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Multistatic Sonobuoy Fields Two tasks of the system: Detect targets that are unknown to the system Accurately track targets known to the system Sensor

Transmitter

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Scheduling Problem # Choose sequence of transmitters and waveforms to satisfy tasks

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Scheduling Problem # Choose sequence of transmitters and waveforms to satisfy tasks At one transmission time: Choose a Transmitter:

T = {j1 , j2 , . . . , jNT }

where NT is the number of transmitters in the field Choose a Waveform:

W = {w1 , w2 , . . . , wNd }

where Nd is the number of possible waveforms

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T = {j1 , j2 , . . . , jNT }


W = {w1 , w2 , . . . , wNd }

where Nd is the number of possible waveforms Possible waveforms: Continuous Wave (CW) or Frequency Modulated (FM) waveform 1kHz or 2kHz frequency 2 second or 8 second duration Gilliam et al.


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T = {j1 , j2 , . . . , jNT }


W = {w1 , w2 , . . . , wNd }

where Nd is the number of possible waveforms Action space: Choose an action:

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a ∈ A,

A=T ×W


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Conflicting Objectives Track vs Search =⇒ Which transmitter to choose...

Sensor

Transmitter

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Conflicting Objectives Our Approach: Combine both tasks in multi-objective framework and use multi-objective optimization to decide scheduling Sensor

Transmitter

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Modelling, Measurements & Tracker

Modelling, Measurements & Tracking Algorithm Sonobuoy Field Description: 60

Transmitter positions T sj = xjs , ysj

Distance (km)

50 40 30

Receiver positions T ri = xir , yri

20 10

Assume positions are known at all times∗

0 0

10

20 30 40 50 Distance (km)

60

‘x’ = Transmitters, ‘o’ = Receivers ∗

Each buoy contains RF communications and may contain GPS equipment Gilliam et al.


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Modelling, Measurements & Tracking Algorithm Target Description: 60

Target Position at time tk : T p = [xk , yk ]

Distance (km)

50 40 30

Target Velocity at time tk : v = [x˙ k , y˙ k ]T

20 10

Time-varying state T xk = [pTk , vTk ]

0 0

10


60

‘x’ = Transmitters, ‘o’ = Receivers

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Modelling, Measurements & Tracking Algorithm Target Motion: 60

Noisy linear constant-velocity model 1 T xk = ⊗ I2 xk−1 + ek 0 1 {z } |

Distance (km)

50 40 30 20

f (xk−1 )

10

Process noise ek is Gaussian with variance 3 T /3 T 2 /2 Q=ω 2 ⊗ I2 T /2 T

0 0

10


60


where T = tk − tk−1 is the sampling in time ⊗ is the Kronecker product and I2 is 2 × 2 identity matrix Gilliam et al.


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Modelling, Measurements & Tracking Algorithm Measurements: 60

Signal amplitude β and Kinematic measurement z (i) (i) z = hj (xk ) + wj

Distance (km)

50 40 30

Measurements collected from a subset of receivers

20 10

Buoys have two waveform modalities

0 0

10


60

Frequency Modulated (FM) Continuous Wave (CW)


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Modelling, Measurements & Tracking Algorithm Using FM waveforms: 60

Bistatic Range: |pk − ri | + |pk − sj |

Distance (km)

50 40 30

Angle from Receiver: y −y i arctan xkk −xri

20

r

10

Good positional information

0 0

10


60


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Modelling, Measurements & Tracking Algorithm Using CW waveforms: 60

Bistatic Range: |pk − ri | + |pk − sj |

Distance (km)

50 40 30

Angle from Receiver: y −y i arctan xkk −xri

20

r

10

Bistatic Range-Rate: h pk −ri vT |p + k −ri |

0 0

10


60

pk −si |pk −si |

Good velocity information


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i

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Modelling, Measurements & Tracking Algorithm Tracking Challenges: 60

High levels of clutter

Distance (km)

50 40

Non-linear measurements

30

Low probability of detection

20 10 0 0

10


60


Many possible algorithms: ML-PDA, MHT, PMHT, JIPDA, PHD/CPHD, ... etc Gilliam et al.


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Modelling, Measurements & Tracking Algorithm The tracker: 60

Multi-Sensor Bernoulli filter[1] (optimal multi-sensor Bayesian filter for a single target)

Distance (km)

50 40 30

Linear Multi-Target (LMT) Paradigm[2]

20 10

Gaussian mixture model implementation[3]

0 0

10


60

Process FM & CW measurements


[1] B. Ristic el al., ‘A tutorial on Bernoulli filters: Theory, implementation and applications’, IEEE Trans. Signal Process., 2013. [2] D. Muˇsicki and B. La Scala, ‘Multi-Target Tracking in Clutter without Measurement Assignment’, IEEE Trans. Aerosp. Electron. Syst., 2008. [3] B. Ristic el al., ‘Gaussian Mixture Multitarget Multisensor Bernoulli Tracker for Multistatic Sonobuoy Fields’, IET Radar, Sonar & Navig., 2017.

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Overview Track Reward Search Reward

Multi-Objective Framework for choosing Maximising rewards: RSearch (a) ⇒ Reward for searching to detect unknown targets RTrack (a) ⇒ Reward for continued tracking of known targets

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Multi-Objective Framework for choosing Maximising rewards: RSearch (a) ⇒ Reward for searching to detect unknown targets RTrack (a) ⇒ Reward for continued tracking of known targets Combine rewards via convex sum: max {αRTrack (a) + (1 − α)RSearch (a)} a

where α ∈ [0, 1]

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where α ∈ [0, 1] Performance depends on α ⇒ Controls trade-off # Different solutions depending on the value of α

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where α ∈ [0, 1] Pareto Optimality: A point is Pareto optimal if there is no other point that can improve one objective without degrading the other. Problem characterised =⇒ Set of Pareto optimal points # Pareto Frontier Gilliam et al.


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Tracking Reward Sensor Transmitter

Predicted Trajectory

Previous Trajectory Information

Given previous tracking: # Measure the gain in tracking information from action a

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Tracking Reward Sensor Predicted Trajectory

Transmitter


Approximate information matrix: " Single track:

trace JPredict +

X

#

Pdi (a)JiMeasure (a)

i∈R

Trace of only the positional elements of information matrix Pdi (a) Expected probability of detecting track Gilliam et al.


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Predicted Information Matrix: T −1 JPredict = Fk−1 Pk−1 [Fk−1 ] | {z } Propagation of error covariance due to motion model

where Fk−1 is the Jacobian of f (xk−1 ) and Pk−1 is the error covariance from tracker Gilliam et al.


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Measurement Information Matrix: T i −1 i JMeasure = Hik (a) Rk (a) Hk (a) | {z } Gain in information from action

where Hik (a) is the Jacobian of ha (xk−1 ) and Rik (a) is the measurement covariance Gilliam et al.


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Tracking Reward Sensor Predicted Trajectory

Transmitter


Multiple tracks: RTrack (a) =

T X

τ =1

ωτ trace

"

JτPredict

+

X

#

Pdi,τ (a)Ji,τ Measure (a)

i∈R

ωτ ⇒ Normalised weights (∝ 1/existence probability) Gilliam et al.


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Search Reward Reduction of the probability of undetected targets in sonar field

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Search Reward Reduction of the probability of undetected targets in sonar field Modelling this probability[1] : Define Threat Map PT,k ⇒ Discrete 2D grid of probabilities Probabilities evolve over time Increases ⇒ Drift & diffusion of undetected targets Decreases ⇒ Transmitters emits a ping

[1] D. Krout el al., ‘Probability of target presence for multistatic sonar ping sequencing ’, IEEE J. Ocean. Eng., 2009.

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Search Reward Reduction of the probability of undetected targets in sonar field Drift & diffusion process: Matrix G ⇒ Probability of targets entering from adjacent cells Update to Threat Map ⇒ Filter PT,k with G Pre-calculate G using Monte-Carlo simulations e.g. for a 60 s interval, grid size of 1 km, uniformly distributed target speed between 0 and 10 knots   0.0036 0.0582 0.0036 G = 0.0582 0.7526 0.0582 0.0036 0.0582 0.0036

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Search Reward Reduction of the probability of undetected targets in sonar field Transmitting a ping: Apply Bayesian update at each cell of PT,k

PT,k (x, a) =

(1 − Pd (x, a))PT,k−1 (x) (1 − Pd (x, a))PT,k−1 (x) + (1 − Pfa (1 − PT,k−1 (x))

Pd (x, a) is the probability a target is detected after action a Pfa is the false alarm probability x = (x, y) is the 2D grid point

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Search Reward Reduction of the probability of undetected targets in sonar field Obtaining Pd (x, a): Generate probabilities using Monte-Carlo simulations and the realistic simulator (BRISE) 1

e.g.

0.9

160 × 160 km area 1km × 1km grid resolution

0.8 0.7 0.6

5 × 5 transmitter grid 6 × 6 receiver grid

0.5 0.4 0.3

Buoy separation = 15km

0.2

FM, 1 kHz waveform with 2 s duration. Gilliam et al.

0.1 0


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Search Reward Reduction of the probability of undetected targets in sonar field and finally... Rsearch (a) =

X

PT,k−1 (x) − PT,k (x, a)

x

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Analysis of Scheduler - Set Up 60

Set-up:

Distance (km)

50

4 × 4 transmitter grid

40

5 × 5 receiver grid 30


20

50 Minute Scenario

10

1 transmission/minute Blue target present for whole duration

0 0

10


60

Green target appears after 10 minutes

‘x’ = Transmitters, ‘o’ = Receivers Realistic measurements =⇒ Bistatic Range Independent Signal Excess (BRISE) simulation environment Gilliam et al.


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Analysis of Scheduler - Set Up 60

Set-up:

Distance (km)

50

4 × 4 transmitter grid

40

5 × 5 receiver grid 30


20

50 Minute Scenario

10

1 transmission/minute Blue target present for whole duration

0 0

10


60

Green target appears after 10 minutes


Analyse the performance of the scheduler as α varies Gilliam et al.


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Analysis of Scheduler - Demo Undetected Targets

2

Scene Geometry

Threat Map

60

80 1 60

40

Distance (km)

Distance (km)

50

30 20

0

40

10

30

40

50

40

50

Track Error (m) 50

0

10

20

Time (mins) 20 40 30

0

-20 0

10

20

30

40

Distance (km)

50

60

20 -20

0

20

40

60

Distance (km)

80

10 0 10

20

30

Time (mins)

α = 0.35

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Analysis of Scheduler - Demo Undetected Targets

2

Scene Geometry

Threat Map

60

80 1 60

40

Distance (km)

Distance (km)

50

30 20

0

40

10

30

40

50

40

50

Track Error (m) 50

0

10

20

Time (mins) 20 40 30

0

-20 0

10

20

30

40

Distance (km)

50

60

20 -20

0

20

40

60

Distance (km)

80

10 0 10

20

30

Time (mins)

α = 0.35

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Analysis of Scheduler - Results Mean Number of Undetected Targets

12

Mean Track Error (m)

11 10 9 8 7 6 5 4 3

0

0.2

0.4 0.6 Values of α

0.8

1

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3

0

0.2

0.4 0.6 Values of α

0.8

1

Mean Number of Undetected Targets


Error bars = 95% confidence intervals for the estimated values Red dashed line = Performance from random scheduling Values averaged over 300 Monte-Carlo simulations and every transmission Gilliam et al.


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Analysis of Scheduler - Results Pareto-esque Frontier: 12


11 10 9 8 7

Random 0.1 0.2 0.25

0.0

0.3 0.325 0.35 0.355 0.3625

6

0.375 0.4

5 4 3 0.3

0.45 0.5 0.6 0.9 0.8 0.7 1.0

0.4 0.5 0.6 0.7 0.8 0.9 Mean Number of Undetected Targets

1

Values averaged over 300 Monte-Carlo simulations and every transmission Gilliam et al.


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Analysis of Scheduler - Transmitter Choice α=0

40

3 30

2

20

10

1

Proportion of Transmissions (%)

50

4

0

1

2

3

4

2D histogram showing the proportion of waveforms transmitted Gilliam et al.


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Analysis of Scheduler - Transmitter Choice α=0

40

3 30

2

20

10

1

Proportion of Transmissions (%)

50

4

0

1

2

3

4

2D histogram showing the proportion of waveforms transmitted Gilliam et al.


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Conclusions Introduced scheduling of multistatic sonobuoy fields Search =⇒ Detect targets that are unknown Track =⇒ Accurately track known targets

Presented multi-objective framework for scheduling Each task is treated as a separate objective Objectives combined via weighted sum Weight α controls priority placed on each objective

Analysed proposed scheduling via realistic simulations Demonstrated trade-off between search and track as α varies Trade-off characterised in terms of points on the Pareto front

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The End

Thank you for listening

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Scheduling of Multistatic Sonobuoy Fields using Multi

Scheduling of Multistatic Sonobuoy Fields using Multi

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