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1 Position Estimation for Sensor Networks FRC Seminar – Dec. 19, 2007 Joseph Djugash (Speaking Qualifier Talk)

Position Estimation for Sensor Networks

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Position Estimation for Sensor Networks. FRC Seminar – Dec. 19, 2007 Joseph Djugash (Speaking Qualifier Talk). Motivation. Motivation. The Problem. Accurate localization of a large network of nodes. What makes it hard?. Resource Limitation - PowerPoint PPT Presentation

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Page 1: Position Estimation for Sensor Networks

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Position Estimation for Sensor Networks

FRC Seminar – Dec. 19, 2007

Joseph Djugash

(Speaking Qualifier Talk)

Page 2: Position Estimation for Sensor Networks

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Motivation

Page 3: Position Estimation for Sensor Networks

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Motivation

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The Problem

Accurate localization of a large network of nodes

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What makes it hard?o Resource Limitation

o power, communication bandwidth, processing, cost, sensor range, etc.

o Scalability o 10, 100, 1000's of sensor

nodes

o Robustness o maintaining accuracy under

sub-optimal configurations

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Outlineo Range-Only Estimationo Simple Optimizationo Bayesian Estimationo Decentralizationo Conclusion

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Why use range sensors?o Shortcomings of classical

sensorso Line-of-sighto Practical Considerationso Environmental Constraintso Correspondence Problem

o Range-only sensors o Non-Gaussian noise modelso Nonlinear measurements

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Limitations of rangeo Highly nonlinear a measurements

Uncertainty

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Outlineo Range-Only Sensorso Simple Optimization

o The Naïve Approacho Improved Optimization

o Bayesian Estimationo Decentralizationo Conclusion

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Problem Formulationo Inputs:

o Zik: Range meas. btw. nodes i & k

o Outputs:o : Node

positions

o Estimated node positions can be used to predict the input ranges

Page 11: Position Estimation for Sensor Networks

Borg1997, Moore2004, Moses2002 11

Multi-Dimensional Scaling (MDS)

o MDS maps the distances between the nodes into a 2D space.o Minimize,

o Initial condition importanto Invariant to rotation and translation

o To uniquely determine a node’s relative position, it needs to belong to a clique of degree 4 or higher

Observed distances btw nodes i and k

Distances btw nodes within the estimate

Fully connected sub-graph

Page 12: Position Estimation for Sensor Networks

Borg1997, Moore2004, Moses2002 12

Multi-Dimensional Scaling (MDS)

– 1015.811

– –

10 – –15.811

15.811

– – 2014.142

–15.811

20 –14.142

– –14.142

14.142

Ground Truth Positions

Prediction #2

Prediction #1

3 out of 4 meas. needed for rigidity

Page 13: Position Estimation for Sensor Networks

Borg1997, Moore2004, Moses2002 13

Key Problem with MDS

Requires High Degree of Connectivity!

Can we get around this?

Page 14: Position Estimation for Sensor Networks

Kehagias2006, Djugash2006 14

Incorporating Motiono Points along the trajectory are

used to increase the degree of connectivity

o Motion helps resolve ambiguities in orientation and handedness

Sensor RangeMobile Node’s Path

Un-localizable Nodes.

Sufficiently constrained Localizable Nodes.

Robot path samples treated as nodes within the network.

Page 15: Position Estimation for Sensor Networks

Kehagias2006, Djugash2006 15

Improved Optimizationo Minimize the error in …

o All range measurementso Use path history of mobile

nodes to provide additional constraints

o Model noise in measurements

o The Cost Function: Cost for deviating from

robot’s odometryCost of errors in range

measurementsCost

Uncertainty in motion

Uncertainty in measurements

Page 16: Position Estimation for Sensor Networks

Kehagias2006, Djugash2006 16

Shortcomings of Optimization

o Increased DimensionalityoMulti-modality in the

estimate is hidden

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What’s next?

How can we model these ambiguities (uncertainty)

in the estimate?

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Outlineo Range-Only Sensorso Simple Optimizationo Bayesian Estimation

o Bayes Filtero Particle Filtero Parametric Representation

o Decentralizationo Conclusion

Page 19: Position Estimation for Sensor Networks

Thrun2005 19

Bayes Filtero General Formalismo Arbitrary belief

representation

o Recursively computes the posterior distribution:)|()( TTT ZxPxBel

1111 )(),|()( tttttt dxxBelxuxPxBel'

)()|()( tttt xBel'xzPxBel

Motion Model

Sensor Model

Page 20: Position Estimation for Sensor Networks

Thrun2005 20

Bayesian Estimation

Ground Truth Positions

Origin Anchor

Node #2 Node #3 Node #4

Using only meas. from

nodes 1 and 2

Adding angle constraint for Axis Anchor

Axis Anchor

Page 21: Position Estimation for Sensor Networks

Thrun2005 21

Major Drawbacks

o Requires complex belief representation

o Computational costs grow with environment size

o How can we reduce the computational costs?

Page 22: Position Estimation for Sensor Networks

Ihler2004, Ing2005 22

Particle Filteringo Represent belief using a set

of samples or particles

o Sequential importance sampling with re-sampling used to update the belief

o Handles arbitrary motion and measurement model

Page 23: Position Estimation for Sensor Networks

Ihler2004, Ing2005 23

Particle Filtering

True Nodes

Measurements between Nodes

Anchor Nodes

θ

Angle θ determined arbitrarily

Particles

Annulus from 1st Measurement

Current Node Particles

Annulus from 2nd Measurement

Updated Node Particles

Current Node Particles

Annuli from Previous Node

with Two Modes

Final Node Particles

Final Estimate for Local Map

Page 24: Position Estimation for Sensor Networks

Ihler2004, Ing2005 24

Downside to Particle Filters

Poor Scalabilityo Accuracy ∝ (# of Particles) ∝ Computational

Cost

Page 25: Position Estimation for Sensor Networks

Ihler2004, Ing2005 25

Issue of Scalabilityo Consider what happens when a

single additional node is added…

New Node

Page 26: Position Estimation for Sensor Networks

Ihler2004, Ing2005 26

Issue of Scalabilityo Exponential growth of

modeso # of modes ≤ 2 * (# of

modes of observers/“parents”)

o Additional particles needed to accurately represent the nonlinearity within each mode

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How to solve this?o Use of negative information

o Ideal for certain scenarioso Difficult to determine the

cause for lack of info.

o Moving away from particles? Perhaps a more approximate representation of belief?

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Alternate Belief Representations

o How to best approximate the nonlinearity in the belief?

o Idea: Perhaps in a parameterized model this nonlinear distribution will become linear…

o What is a good parameterization?

Page 29: Position Estimation for Sensor Networks

Djugash2008, Funiak2006 29

o Simple Gaussian Parameterization in [x,y] is not sufficient

o Relative Over-Parameterization (ROP)o The ring-like structure can be

represented in polar coordinateso range, theta, center of circle (location

of unknown person) – [r, , mx, my]

Over-Parameterized Filter

True Posterior Gaussian in [x,y]

Page 30: Position Estimation for Sensor Networks

Djugash2008, Funiak2006 30

ROP Representationθ

r

0

θ

r

0

Page 31: Position Estimation for Sensor Networks

Thrun2005, Djugash2008 31

Multi-Modal Distributions

o Standard EKF limited to unimodal Gaussian

o Multiple hypothesis representationo Use multiple EKFs, one for

each hypothesiso Inconsistent hypotheses are

dropped (threshold on likelihood)

Page 32: Position Estimation for Sensor Networks

Djugash2008 32

Example of ROP-EKF

Page 33: Position Estimation for Sensor Networks

Djugash2008, Funiak2006 33

Drawbacks of ROP-EKF

o Accuracy limited by parameterization

o Singularities requires special consideration

o Hypothesis count limits scalability

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Outlineo Range-Only Sensorso Simple Optimizationo Bayesian Estimationo Decentralizationo Conclusion

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DecentralizationoHow to distribute the

work load without sacrificing accuracy?oCan we guarantee …

orobustness? oconvergence?

oWhat, if any, information needs to be shared?

Page 36: Position Estimation for Sensor Networks

Sudderth2003 36

Belief Propagationo An Inference method on graphso The set of sensor nodes are the graphical

modelo Combine the observations from all nodes via

message-passing operations

o Belief Computation

Normalization Constant

Belief = of all inputs into node “s”

Observations of node “s”

Messages from neighbors

Page 37: Position Estimation for Sensor Networks

Sudderth2003 37

Belief Propagationo Message Computation

o Message Product:o Belief based on all nodes except node “s”

o Message Propagation:o Marginalize over node “t” to compute

belief of node “s”

Message ProductMessage

Propagation

Page 38: Position Estimation for Sensor Networks

Sudderth2003, Ihler2004 38

Properties of BPo Produces exact conditional

marginals for tree-like graphs

o Excellent empirical performance

o Nonparametric BP – Ihler2004

o Non-Gaussian and continuous distributions

o Transmit samples of the message distribution

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Outlineo Range-Only Sensorso Simple Optimizationo Bayesian Estimationo Decentralizationo Conclusion

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Comparison

AccuracyRobustne

ss

Computation

{Low - High}

Scalability

{10 - 1000}

Communication

{Low – High}

MDS 1 1 Low 1000’s Low

Optim. w/ Motion

3 2 High 10’s Med.

Full Bayes Filter

5 5 High <10’s Med.

Particle Filter

4 4 Med. 10’s Med.

ROP EKF 3 3 Low – Med. 100’s Med.

ROP EKF w/ BP

3 3 Low >100’s Low – Med.

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Complexity vs. Accuracy

o Striking a Good Compromise Requireso Improved Representation!o Distributable Computation!

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Referenceso Borg1997: I. Borg and P. Groenen, “Modern multidimensional

scaling: theory and applications,” New York: Springer, 1997.

o Moore2004: D. Moore, J. Leonard, D. Rus, and S. Teller, “Robust distributednetwork localization with noisy range measurements,” in in Sen-Sys’04: Proc 2nd international conference on Embedded networked sensor systems. New York: ACM Press, 2004, pp. 50–61.

o Moses2002: R. Moses and R. Patterson, “Self-calibration of sensor networks,” Unattended Ground Sensor Technologies and Applications IV, vol. 4743 in SPIE, 2002.

o Kehagias2006: A. Kehagias, J. Djugash, and S. Singh, “Range-only slam with interpolated range data,” tech. report CMU-RI-TR-06-26, Robotics Institute, Carnegie Mellon University, May, 2006, Tech. Rep.

o Djugash2006: J. Djugash, S. Singh, G. Kantor, and W. Zhang, “Range-only slam for robots operating cooperatively with sensor networks,” in IEEE Int’l Conf. on Robotics and Automation (ICRA ‘06), 2006.

o Thrun2005: S. Thrun, W. Burgard, and D. Fox, Probabilistic Robotics. Cambridge, MA: MIT Press, 2005.

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Referenceso Ihler2004: A. T. Ihler, J. W. Fisher III, R. L. Moses, and A. S.

Willsky, “Nonparametric belief propagation for self-calibration in sensor networks,” in Information Processing in Sensor Networks, 2004.

o Ing2005: G. Ing, M.J.Coates, "Parallel particle filters for tracking in wireless sensor networks," Signal Processing Advances in Wireless Communications, 2005 IEEE 6th Workshop on , vol., no., pp. 935-939, 5-8 June 2005

o Funiak2006: S. Funiak, C. E. Guestrin, R. Sukthankar, and M. Paskin, “Distributed localization of networked cameras,” in Fifth International Conference on Information Processing in Sensor Networks (IPSN’06), April 2006, pp. 34 – 42.

o Stump2006: E. Stump, B. Grocholsky, and V. Kumar, “Extensive representations and algorithms for nonlinear filtering and estimation,” in The Seventh International Workshop on the Algorithmic Foundations of Robotics, July 2006.

o Djugash2008: J. Djugash, B. Grocholsky, and S. Singh, “Decentralized Mapping of Robot-Aided Sensor Networks,” in IEEE Int’l Conf. on Robotics and Automation (ICRA ‘08), 2008.

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Referenceso Sudderth2003: E. Sudderth, A. Ihler, W. Freeman, and A.

Willsk, “Nonparametric Belief Propagation,” Computer Vision and Pattern Recognition (CVPR), June 2003.

o Olfati-Saber2005: R.Olfati-Saber, J.S.Shamma, "Consensus Filters for Sensor Networks and Distributed Sensor Fusion," Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC '05. 44th IEEE Conference on , vol., no., pp. 6698-6703, 12-15 Dec. 2005

o Paskin2005: M. Paskin, C. Guestrin, and J. McFadden. “A robust architecture for inference in sensor networks,” In Proc. IPSN, 2005.

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Thank You

Advisor: Sanjiv Singh

Committee MembersBrett Browning

Paul RybskiNathaniel Fairfield

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Conclusiono Motion helps with sparse

connectivityo Modeling of uncertainty is

necessary o Parametric belief

representations o Preserve scalability and

robustnesso Little loss in accuracy

o Decentralization improves scalability

Page 48: Position Estimation for Sensor Networks

Djugash2008 48

Belief Propagation with ROP-EKF

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Exploiting Negative Information

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Coordinate System + Handednesso In the absence of anchor nodes…

o Arbitrarily assign a node to the origin o A second node (observable from the origin

node) determines one of the axiso The other axis is left ambiguouso Unless handedness is resolved, the flip

solution offers another equally likely solution in most cases

Global Coordinate

Z = range btw node

Z

Estimate Coordinate

Z

One Solution

Estimate Coordinate

Z

Flip Solution

Origin Anchor Axis

Anchor