Position Estimation for Sensor Networks

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Position Estimation for Sensor Networks

• FRC Seminar Dec. 19, 2007
• Joseph Djugash
• (Speaking Qualifier Talk)

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Motivation
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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?

• Resource Limitation
• power, communication bandwidth, processing, cost,
  sensor range, etc.
• Scalability
• 10, 100, 1000's of sensor nodes
• Robustness
• maintaining accuracy under sub-optimal
  configurations

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Outline

• Range-Only Estimation
• Simple Optimization
• Bayesian Estimation
• Decentralization
• Conclusion

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Why use range sensors?

• Shortcomings of classical sensors
• Line-of-sight
• Practical Considerations
• Environmental Constraints
• Correspondence Problem
• Range-only sensors
• Non-Gaussian noise models
• Nonlinear measurements

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Limitations of range

• Highly nonlinear a measurements

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Outline

• Range-Only Sensors
• Simple Optimization
• The Naïve Approach
• Improved Optimization
• Bayesian Estimation
• Decentralization
• Conclusion

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Problem Formulation

• Inputs
• Zik Range meas. btw. nodes i k
• Outputs
• Node
  positions
• Estimated node positions can be used to predict
  the input ranges

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Multi-Dimensional Scaling (MDS)

• MDS maps the distances between the nodes into a
  2D space.
• Minimize,
• Initial condition important
• Invariant to rotation and translation
• To uniquely determine a nodes 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
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Multi-Dimensional Scaling (MDS)
3 out of 4 meas. needed for rigidity
Ground Truth Positions
10 15.811
10 15.811
15.811 20 14.142
15.811 20 14.142
14.142 14.142
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Key Problem with MDS

• Requires High Degree of Connectivity!
• Can we get around this?

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Incorporating Motion

• Points along the trajectory are used to increase
  the degree of connectivity
• Motion helps resolve ambiguities in orientation
  and handedness

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Improved Optimization

• Minimize the error in
• All range measurements
• Use path history of mobile nodes to provide
  additional constraints
• Model noise in measurements
• The Cost Function

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Shortcomings of Optimization

• Increased Dimensionality
• Multi-modality in the estimate is hidden

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Whats next?

• How can we model these ambiguities (uncertainty)
  in the estimate?

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Outline

• Range-Only Sensors
• Simple Optimization
• Bayesian Estimation
• Bayes Filter
• Particle Filter
• Parametric Representation
• Decentralization
• Conclusion

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Bayes Filter

• General Formalism
• Arbitrary belief representation
• Recursively computes the posterior distribution

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Bayesian Estimation
Using only meas. from nodes 1 and 2
Origin Anchor
Adding angle constraint for Axis Anchor
Ground Truth Positions
Node 2
Node 3
Node 4
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Major Drawbacks

• Requires complex belief representation
• Computational costs grow with environment size
• How can we reduce the computational costs?

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Particle Filtering

• Represent belief using a set of samples or
  particles
• Sequential importance sampling with re-sampling
  used to update the belief
• Handles arbitrary motion and measurement model

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Particle Filtering
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Downside to Particle Filters

• Poor Scalability
• Accuracy ? ( of Particles) ? Computational Cost

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Issue of Scalability

• Consider what happens when a single additional
  node is added

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Issue of Scalability

• Exponential growth of modes
• of modes 2 ? ( of modes of
  observers/parents)
• Additional particles needed to accurately
  represent the nonlinearity within each mode

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How to solve this?

• Use of negative information
• Ideal for certain scenarios
• Difficult to determine the cause for lack of
  info.
• Moving away from particles? Perhaps a more
  approximate representation of belief?

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Alternate Belief Representations

• How to best approximate the nonlinearity in the
  belief?
• Idea Perhaps in a parameterized model this
  nonlinear distribution will become linear
• What is a good parameterization?

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Over-Parameterized Filter

• Simple Gaussian Parameterization in x,y is not
  sufficient
• Relative Over-Parameterization (ROP)
• The ring-like structure can be represented in
  polar coordinates
• range, theta, center of circle (location of
  unknown person) r, ?, mx, my

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ROP Representation
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Multi-Modal Distributions

• Standard EKF limited to unimodal Gaussian
• Multiple hypothesis representation
• Use multiple EKFs, one for each hypothesis
• Inconsistent hypotheses are dropped (threshold on
  likelihood)

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Example of ROP-EKF
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Drawbacks of ROP-EKF

• Accuracy limited by parameterization
• Singularities requires special consideration
• Hypothesis count limits scalability

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Outline

• Range-Only Sensors
• Simple Optimization
• Bayesian Estimation
• Decentralization
• Conclusion

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Decentralization

• How to distribute the work load without
  sacrificing accuracy?
• Can we guarantee
• robustness?
• convergence?
• What, if any, information needs to be shared?

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Belief Propagation

• An Inference method on graphs
• The set of sensor nodes are the graphical model
• Combine the observations from all nodes via
  message-passing operations
• Belief Computation

Observations of node s
Messages from neighbors
Belief ? of all inputs into node s
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Belief Propagation

• Message Computation
• Message Product
• Belief based on all nodes except node s
• Message Propagation
• Marginalize over node t to compute belief of
  node s

Message Product
Message Propagation
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Properties of BP

• Produces exact conditional marginals for
  tree-like graphs
• Excellent empirical performance
• Nonparametric BP Ihler2004
• Non-Gaussian and continuous distributions
• Transmit samples of the message distribution

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Outline

• Range-Only Sensors
• Simple Optimization
• Bayesian Estimation
• Decentralization
• Conclusion

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Comparison
Accuracy Robustness ComputationLow - High Scalability10 - 1000 CommunicationLow High
MDS 1 1 Low 1000s Low
Optim. w/ Motion 3 2 High 10s Med.
Full Bayes Filter 5 5 High lt10s Med.
Particle Filter 4 4 Med. 10s Med.
ROP EKF 3 3 Low Med. 100s Med.
ROP EKF w/ BP 3 3 Low gt100s Low Med.
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Complexity vs. Accuracy

• Striking a Good Compromise Requires
• Improved Representation!
• Distributable Computation!

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References

• Borg1997 I. Borg and P. Groenen, Modern
  multidimensional scaling theory and
  applications, New York Springer, 1997.
• Moore2004 D. Moore, J. Leonard, D. Rus, and S.
  Teller, Robust distributednetwork localization
  with noisy range measurements, in in Sen-Sys04
  Proc 2nd international conference on Embedded
  networked sensor systems. New York ACM Press,
  2004, pp. 5061.
• Moses2002 R. Moses and R. Patterson,
  Self-calibration of sensor networks, Unattended
  Ground Sensor Technologies and Applications IV,
  vol. 4743 in SPIE, 2002.
• 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.
• Djugash2006 J. Djugash, S. Singh, G. Kantor, and
  W. Zhang, Range-only slam for robots operating
  cooperatively with sensor networks, in IEEE
  Intl Conf. on Robotics and Automation (ICRA
  06), 2006.
• Thrun2005 S. Thrun, W. Burgard, and D. Fox,
  Probabilistic Robotics. Cambridge, MA MIT Press,
  2005.

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References

• 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.
• 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
• 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 (IPSN06), April
  2006, pp. 34 42.
• 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.
• Djugash2008 J. Djugash, B. Grocholsky, and S.
  Singh, Decentralized Mapping of Robot-Aided
  Sensor Networks, in IEEE Intl Conf. on Robotics
  and Automation (ICRA 08), 2008.

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References

• Sudderth2003 E. Sudderth, A. Ihler, W. Freeman,
  and A. Willsk, Nonparametric Belief
  Propagation, Computer Vision and Pattern
  Recognition (CVPR), June 2003.
• 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
• 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 Members
• Brett Browning
• Paul Rybski
• Nathaniel Fairfield

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(No Transcript)
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Conclusion

• Motion helps with sparse connectivity
• Modeling of uncertainty is necessary
• Parametric belief representations
• Preserve scalability and robustness
• Little loss in accuracy
• Decentralization improves scalability

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Belief Propagation with ROP-EKF
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Exploiting Negative Information
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Coordinate System Handedness

• In the absence of anchor nodes
• Arbitrarily assign a node to the origin
• A second node (observable from the origin node)
  determines one of the axis
• The other axis is left ambiguous
• Unless handedness is resolved, the flip solution
  offers another equally likely solution in most
  cases

Z range btw node
One Solution
Flip Solution
Z
Z
Z
Origin Anchor
Axis Anchor
Global Coordinate
Estimate Coordinate
Estimate Coordinate