1 position estimation for sensor networks frc seminar – dec. 19, 2007 joseph djugash (speaking...
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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?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
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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
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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
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Borg1997, Moore2004, Moses2002 13
Key Problem with MDS
Requires High Degree of Connectivity!
Can we get around this?
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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.
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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
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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
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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
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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
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Thrun2005 21
Major Drawbacks
o Requires complex belief representation
o Computational costs grow with environment size
o How can we reduce the computational costs?
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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
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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
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Ihler2004, Ing2005 24
Downside to Particle Filters
Poor Scalabilityo Accuracy ∝ (# of Particles) ∝ Computational
Cost
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Ihler2004, Ing2005 25
Issue of Scalabilityo Consider what happens when a
single additional node is added…
New Node
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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?
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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]
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Djugash2008, Funiak2006 30
ROP Representationθ
r
2π
0
θ
r
2π
0
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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)
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Djugash2008 32
Example of ROP-EKF
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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?
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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
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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
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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
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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