decentralised coordination of mobile sensors using the max-sum algorithm ruben stranders, alex...
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![Page 1: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/1.jpg)
Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm
Ruben Stranders, Alex Rogers, Nick JenningsSchool of Electronics and Computer ScienceUniversity of Southampton{rs06r, acr, nrj}@ecs.soton.ac.uk
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2
This presentation focuses on the use of Max-Sum to coordinate mobile sensors
Sensor Architecture & Max-Sum
Empirical Evaluation
Speeding up Max-Sum
Model
Value
Coordinate
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This work can be applied to improve situational awareness in dynamic scenarios
Disaster Response
Military Surveillance
Climate Research
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Our contribution is a coordination mechanism for a team of autonomous mobile sensors
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These mobile sensors continuously monitor spatial phenomena
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The main challenge is to coordinate the sensors in order to the state of these spatial phenomena
![Page 7: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/7.jpg)
The main challenge is to coordinate the sensors in order to the state of these spatial phenomena
?
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The main challenge is to coordinate the sensors in order to the state of these spatial phenomena
Limited Communication
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The main challenge is to coordinate the sensors in order to the state of these spatial phenomena
No centralised control
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The main challenge is to coordinate the sensors in order to the state of these spatial phenomena
No centralised control
![Page 11: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/11.jpg)
To solve this coordination problem, we had to address three challenges
1. How to model the phenomena?2. How to value potential samples?3. How to coordinate to gather
samples of highest value?
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The three central challenges are clearly reflected in the architecture of our sensing agents
Samples sent toneighbouring agents
Samples received fromneighbouring agents
Information processing
Model of Environment
Outgoing negotiation messages
Incomingnegotiation messages
Value of potential samples Action
Selection
Move
Samples from own sensor
SensingAgent
Rawsamples
Model
Value
Coordinate
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These three challenges are clearly reflected in the architecture of our sensing agents
Samples sent toneighbouring agents
Samples received fromneighbouring agents
Information processing
Model of Environment
Outgoing negotiation messages
Incomingnegotiation messages
Value of potential samples Action
Selection
Move
Samples from own sensor
SensingAgent
Rawsamples
Model
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The sensors model the spatial phenomenon using the Gaussian Process
Weak Strong
Spatial Correlations
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The sensors model the spatial phenomenon using the Gaussian Process
Areas of Rapid Change
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The sensors model the spatial phenomenon using the Gaussian Process
Weak Strong
Temporal Correlations
0 0.5 1 1.5 2 2.5 3 3.5 4
-2
Time
Tem
pera
ture
0 0.5 1 1.5 2 2.5 3 3.5 4
-2
Time
Tem
pera
ture
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The value of a sample is determined how much it reduces uncertainty
Samples sent toneighbouring agents
Samples received fromneighbouring agents
Information processing
Model of Environment
Outgoing negotiation messages
Incomingnegotiation messages
Value of potential samples Action
Selection
Move
Samples from own sensor
SensingAgent
Rawsamples
Value
![Page 18: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/18.jpg)
The value of a sample is based on how much it reduces uncertainty
But how to determine uncertainty reduction before collecting a sample?
![Page 19: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/19.jpg)
The value of a sample is based on how much it reduces uncertainty
But how to determine uncertainty reduction before collecting a sample?
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
PredictionConfidence IntervalCollected Sample
Gaussian Process not only gives predictions, but also confidence intervals
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The value of a sample is based on how much it reduces uncertainty
But how to determine uncertainty reduction before collecting a sample?
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
PredictionConfidence IntervalCollected Sample
Gaussian Process not only gives predictions, but also confidence intervals
Potential Sample Location
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The value of a sample is based on how much it reduces uncertainty
But how to determine uncertainty reduction before collecting a sample?
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
PredictionConfidence IntervalCollected Sample
Gaussian Process not only gives predictions, but also confidence intervals
Measure of uncertainty
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The value of a sample is based on how much it reduces uncertainty
0 1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
PredictionConfidence IntervalCollected Sample
Specifically, we can use information metrics such as Entropy ,or Mutual Information
)(XH
);( YXMI
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The sensor agents coordinate using the Max-Sum algorithm
Samples sent toneighbouring agents
Samples received fromneighbouring agents
Information processing
Model of Environment
Outgoing negotiation messages
Incomingnegotiation messages
Value of potential samples Action
Selection
Move
Samples from own sensor
SensingAgent
Rawsamples
Coordinate
![Page 24: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/24.jpg)
24
Max-Sum is a powerful algorithm for solving DCOPs
Complete Algorithms
DPOPOptAPOADOPT
Communication Cost
Iterative Algorithms
Best Response (BR)Distributed Stochastic
Algorithm (DSA) Fictitious Play (FP)
Max-SumAlgorithm
Optimality
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Max-Sum solves the social welfare maximisation problem in a decentralised way
Mobile Sensors
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Max-Sum solves the social welfare maximisation problem in a decentralised way
1x
2x
3x
4x
5x
6x
7x8x
Movement Parameters
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Max-Sum solves the social welfare maximisation problem in a decentralised way
1U
2U
3U
4U
5U
6U
7U8U
Utility Functions
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Max-Sum solves the social welfare maximisation problem in a decentralised way
)( 33 xU
Localised Interaction
},,,{ 54313 xxxxx
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Max-Sum solves the social welfare maximisation problem in a decentralised way
M
iiiU
1
)(maxarg xx
Social welfare:
Mobile Sensors
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The input for the Max-Sum algorithm is a graphical representation of the problem: a Factor Graph
Variable nodes Function nodes
1x
2x
3x
1U
2U
3U
Agent 1Agent 2
Agent 3
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Max-Sum solves the social welfare maximisation problem by message passing
1x
2x
3x
1U
2U
3U
Variable nodes Function nodes
Agent 1Agent 2
Agent 3
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Max-Sum solves the social welfare maximisation problem by message passing
jiadjk
iikiji xrxq\)(
)()(
ijadjk
kjkjji
iij xqUxrj \)(\
)()(max)( xx
From variable i to function j
From function j to variable i
![Page 33: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/33.jpg)
To use Max-Sum, we encode the mobile sensor coordination problem as a factor graph
1x
2x
3x
1U
2U
3U
Sensor 1Sensor 2
Sensor 3
Sensor 1
Sensor 2
Sensor 3
![Page 34: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/34.jpg)
Variables represent the sensors’ movements
1x
2x
3x
1U
2U
3U
Sensor 1Sensor 2
Sensor 3
![Page 35: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/35.jpg)
Functions represent the uncertainty reduction that results from collecting a sample
1x
2x
3x
1U
2U
3U
Sensor 1Sensor 2
Sensor 3
![Page 36: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/36.jpg)
Functions represent the uncertainty reduction that results from collecting a sample
1x
2x
3x
1U
2U
3U
Sensor 1Sensor 2
Sensor 3
![Page 37: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/37.jpg)
Functions represent the uncertainty reduction that results from collecting a sample
1x
2x
3x
1U
2U
3U
Sensor 1Sensor 2
Sensor 3
![Page 38: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/38.jpg)
Functions represent the uncertainty reduction that results from collecting a sample
1x
2x
3x
1U
2U
3U
Sensor 1Sensor 2
Sensor 3
![Page 39: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/39.jpg)
Functions represent the uncertainty reduction that results from collecting a sample
1x
2x
3x
1U
2U
3U
Sensor 1Sensor 2
Sensor 3
![Page 40: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/40.jpg)
ijadjk
kjkjji
iij xqUxrj \)(\
)()(max)( xx
Unfortunately, the straightforward application of Max-Sum is too computationally expensive
jiadjk
iikiji xrxq\)(
)()(
From variable i to function j
From function j to variable i
![Page 41: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/41.jpg)
ijadjk
kjkjji
iij xqUxrj \)(\
)()(max)( xx
Unfortunately, the straightforward application of Max-Sum is too computationally expensive
jiadjk
iikiji xrxq\)(
)()(
From variable i to function j
From function j to variable i
Bottleneck!
![Page 42: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/42.jpg)
ijadjk
kjkjji
iij xqUxrj \)(\
)()(max)( xx
Therefore, we developed two general pruning techniques that speed up Max-Sum
Goal: Make as small as possible
![Page 43: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/43.jpg)
ijadjk
kjkjji
iij xqUxrj \)(\
)()(max)( xx
Therefore, we developed two general pruning techniques that speed up Max-Sum
Goal: Make as small as possible
1. Try to prune the action spaces of individual sensors
2. Try to prune joint actions
ix
ij \x
![Page 44: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/44.jpg)
The first pruning technique prunes individual actions by identifying dominated actions
![Page 45: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/45.jpg)
The first pruning technique prunes individual actions by identifying dominated actions
1. Neighbours send bounds
↑ [2, 2]↓ [1, 1]
↑ [5, 6]↓ [0, 1]
↑ [1, 2]↓ [3, 4]
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The first pruning technique prunes individual actions by identifying dominated actions
2. Bounds are summed[8, 10]
[4, 7]
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The first pruning technique prunes individual actions by identifying dominated actions
3. Dominated actions are pruned [8, 10]
[4, 7]X
![Page 48: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/48.jpg)
ijadjk
kjkjji
iij xqUxrj \)(\
)()(max)( xx
We developed two general pruning techniques that speed up Max-Sum
Goal: Make as small as possible
1. Try to prune the action spaces of individual sensors
2. Try to prune joint actions
ix
ij \x
![Page 49: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/49.jpg)
The second pruning technique reduces the joint action space because exhaustive enumeration is too costly
Action
Sensor 1 Sensor 2 Sensor 3
![Page 50: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/50.jpg)
Action
The second pruning technique reduces the joint action space because exhaustive enumeration is too costly
Sensor 1 Sensor 2 Sensor 3
![Page 51: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/51.jpg)
Action
The second pruning technique reduces the joint action space because exhaustive enumeration is too costly
Sensor 1 Sensor 2 Sensor 3
![Page 52: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/52.jpg)
Action
The second pruning technique reduces the joint action space because exhaustive enumeration is too costly
Sensor 1 Sensor 2 Sensor 3
![Page 53: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/53.jpg)
Action
The second pruning technique reduces the joint action space because exhaustive enumeration is too costly
Sensor 1 Sensor 2 Sensor 3
![Page 54: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/54.jpg)
Action
The second pruning technique reduces the joint action space because exhaustive enumeration is too costly
Sensor 1 Sensor 2 Sensor 3
![Page 55: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/55.jpg)
Action
The second pruning technique reduces the joint action space because exhaustive enumeration is too costly
Sensor 1 Sensor 2 Sensor 3
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Action
The second pruning technique reduces the joint action space because exhaustive enumeration is too costly
Sensor 1 Sensor 2 Sensor 3
Etcetera…
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The second pruning technique prunes the joint action space using branch and bound
Sensor 1
Sensor 2
Sensor 3
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The second pruning technique prunes the joint action space using branch and bound
[7, 13][0, 4] [2, 6]
Sensor 1
Sensor 2
Sensor 3
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The second pruning technique prunes the joint action space using branch and bound
[7, 13][0, 4] [2, 6]XX
Sensor 1
Sensor 2
Sensor 3
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The second pruning technique prunes the joint action space using branch and bound
9 10 7 8
[7, 13][0, 4] [2, 6]XX
Sensor 1
Sensor 2
Sensor 3
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The second pruning technique prunes the joint action space using branch and bound
9 10 7 8
[7, 13][0, 4] [2, 6]XX
X X XO
Sensor 1
Sensor 2
Sensor 3
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This demonstration shows four sensors monitoring a spatial phenomenon
![Page 63: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/63.jpg)
This demonstration shows four sensors monitoring a spatial phenomenon
Sensors
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This demonstration shows four sensors monitoring a spatial phenomenon
UncertaintyContours
![Page 65: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/65.jpg)
This demonstration shows four sensors monitoring a spatial phenomenon
![Page 66: Decentralised Coordination of Mobile Sensors using the Max-Sum Algorithm Ruben Stranders, Alex Rogers, Nick Jennings School of Electronics and Computer](https://reader036.vdocuments.us/reader036/viewer/2022081519/56649d2d5503460f94a03f6b/html5/thumbnails/66.jpg)
To empirically evaluate our algorithm, we measured speed up and prediction error
UncertaintyContours
9 10 7 8
[7, 13][0, 4] [2, 6]XX
X X X
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The two pruning techniques combined prune 95% of the action space with 6 neighbouring sensors
2 2.5 3 3.5 4 4.5 5 5.5 60
25
50
75
100
Number of neighbouring sensors
% o
f joi
nt a
ction
s pr
uned
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Aver
age
RMSE
Our Algorithm reduces Root Mean Squared Error of predictions up to 50% compared to Greedy
Our Al-gorithm
Greedy Random Fixed0.0
0.2
0.4
0.6
0.8
1.0
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In conclusion, the use of Max-Sum leads to an effective coordination algorithm for mobile sensors
1. Decentralised
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In conclusion, the use of Max-Sum leads to an effective coordination algorithm for mobile sensors
1. Decentralised
2. Fast
% P
rune
d
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In conclusion, the use of Max-Sum leads to an effective coordination algorithm for mobile sensors
1. Decentralised
2. Fast
3. Accurate predictions
% P
rune
d
Pred
ictio
n Er
ror
Our
Greedy
Random
Fixed
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For future work, we wish to extend the algorithm to do non-myopic planning
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References
• R. Stranders, A. Farinelli, A. Rogers and N.R. Jennings (2009): Decentralised Coordination of Mobile Sensors Using the Max-Sum Algorithm. In: Proc 21st Int. Joint Conf on AI (IJCAI), Pasadena, USA. (In Press)
• R. Stranders, A. Farinelli, A. Rogers and N.R. Jennings (2009): Decentralised Coordination of Continuously Valued Control Parameters using the Max-Sum Algorithm. 8th Proc. Int. Conf. on Autonomous Agents and Multiagent Systems (AAMAS), Budapest. (In Press)
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In conclusion, the use of Max-Sum leads to an effective coordination algorithm for mobile sensors
1. Decentralised
2. Fast
3. Accurate predictions
% P
rune
d
Pred
ictio
n Er
ror
Our
Greedy
Random
Fixed
Questions?