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IEEE SECON 2004
Estimation Bounds for Localization
October 7th, 2004
Cheng Chang
EECS Dept ,UC Berkeley [email protected]
Joint work with Prof. Anant Sahai
(part of BWRC-UWB project funded by the NSF)
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Outline
Introduction– Range-based Localization as an Estimation Problem
– Cramer-Rao Bounds (CRB)
Estimation Bounds on Localization– Properties of CRB on range-based localization
– Anchored Localization (3 or more nodes with known positions) Lower Bounds Upper Bounds
– Anchor-free Localization
– Different Propagation Models
Conclusions
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Localization Overview
What is localization?– Determine positions of the nodes (relative or absolute)
Why is localization important?– Routing, sensing etc
Information available– Connectivity
– Euclidean distances and angles
– Euclidean distances (ranging) only
Why are bounds interesting– Local computable
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Range-based localization
Range-based Localization– Positions of nodes in set F (anchors, beacons) are known, positions of
nodes in set S are unknown
– Inter-node distances are known among some neighbors
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Localization is an estimation problem
Knowledge of anchor positions
Range observations:– adj(i) = set of all neighbor nodes of node I
– di,j : distance measurement between node i and j
– di,j = di,j true+ ni,j (ni,j is modeled as iid Gaussian throughout most of the talk)
Parameters to be estimated :
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Anchored vs Anchor-free
Anchored localization ( absolute coordinates)– 3 or more anchors are needed
– The positions of all the nodes can be determined.
Anchor-free localization (relative coordinates)– No anchors needed
– Only inter-node distance measurements are available.
– If θ={(xi ,yi)T| i є S} is a parameter vector, θ*={R(xi ,yi)T+(a,b) | i ∊ S} is an equivalent parameter vector, where RRT =I2 .
Performance evaluation– Anchored: Squared error for individual nodes
– Anchor-free: Total squared error
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Fisher Information and the Cramer-Rao Bound
Fisher Information Matrix (FIM)– Fisher Information Matrix (FIM) J provides a tool to compute the best
possible performance of all unbiased estimators
– Anchored: FIM is usually non-singular.
– Anchor-free: FIM is always singular (Moses and Patterson’02)
Cramer-Rao Lower bound (CRB)– For any unbiased estimator :
, ( ) cov log ( | ), log ( | )i ji j
J p x p x
1)())ˆ)(ˆ(( JE T
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Outline
Introduction– Range-based Localization as an Estimation Problem
– Cramer-Rao Bounds (CRB)
Estimation Bounds on Localization– Properties of CRB on range-based localization
– Anchored Localization (3 or more nodes with known positions) Lower Bounds Upper Bounds
– Anchor-free Localization
– Different Propagation Models
Conclusions and Future Work
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FIM for localization (a geometric interpretation)
FIM of the Localization Problem (anchored and anchor-free)– ni,j are modeled as iid Gaussian ~N(0,σ2).
– Let θ=(x1,y1,…xm,ym) be the parameter vector, 2m parameters.
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The standard Cramer-Rao bound analysis works. (FIM nonsingular in general)
– V(xi)=J(θ) -1 2i-1,2i-1 and V(yi) =J(θ) -1 2i,2i are the Cramer-Rao bound on the coordinate-estimation of the i th node.
– CRB is not local because of the inversion.
Translation, rotation and zooming, do not change the bounds.
– J(θ)= J(θ*) , if (x*i,y*i)=(xi ,yi)+(Tx ,Ty)
– V(xi)+V(yi)=V(x*i)+V(y*i), if (x*i,y*i)=(xi ,yi)R, where RRT=I2
– J(θ)= J(aθ) , if a ≠ 0
Properties of CRB for Anchored Localization
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Outline
Introduction– Range-based Localization as an Estimation Problem
– Cramer-Rao Bounds (CRB)
Estimation Bounds on Localization– Properties of CRB on range-based localization
– Anchored Localization (3 or more nodes with known positions) Lower Bounds Upper Bounds
– Anchor-free Localization
– Different Propagation Models
Conclusions
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Local lower bounds: how good can you do?
A lower bound on the Cramer-Rao bound , write θl =(xl ,yl)
Jl is a 2×2 sub-matrix of J(θ) . Then for any unbiased estimator , E(( -θl) T( -θl))≥ Jl
-1
Jl only depends on (xl ,yl) and (xi ,yi) , i ∊ adj(l) so we can give a performance bound on the estimation of (xl ,yl) using only the geometries of sensor l 's neighbors.
Sensor l has W neighbors (W=|adj(l)|), then
ll l
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Lower bound: how good can you do?
Jl is the FIM of another estimation problem of (xl ,yl): knowing the positions of all neighbors and inter-node distance measurements between node l and its neighbors.
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Lower bound: how good can you do?
Jl is the FIM of another estimation problem of (xl ,yl): knowing the positions of all neighbors and inter-node distance measurements between node l and its neighbors.
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Lower bound: how good can you do?
Jl is the ‘one-hop’ sub-matrix of J(θ), Using multiple-hop sub-matrices , we can get tighter bounds.(figure out the computations of it)
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Upper Bound: what’s the best you can do with local information.
An upper bound on the Cramer-Rao bound .– Using partial information can only make the estimation less
accurate.
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Upper Bound: what’s the best you can do with local information.
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Outline
Introduction– Range-based Localization as an Estimation Problem
– Cramer-Rao Bounds (CRB)
Estimation Bounds on Localization– Properties of CRB on range-based localization
– Anchored Localization (3 or more nodes with known positions) Lower Bounds Upper Bounds
– Anchor-free Localization
– Different Propagation Models
Conclusions
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Equivalent class in the anchor-free localization
If α={(xi ,yi)T| i є S} , β={R(xi ,yi)T+(a,b) | i ∊ S} is equivalent to α, where RRT =I2 .– Same inter-node disances
A parameter vector θα is an equivalent class θα ={β |β={R(xi ,yi)T+(a,b) | i ∊ S} }
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Estimation Bound on Anchor-free Localization
The Fisher Information Matrix J(θ) is singular (Moses and Patterson’02)
m nodes with unknown position:
– J(θ) has rank 2m-3 in general
– J(θ) has 2m-3 positive eigenvalues λi, i=1,…2m-3, and they are invariant under rotation, translation and zooming on the whole sensor network.
The error between θ and is defined as
Total estimation bounds
2
2',inf)ˆ,( D
)(1
))ˆ,((32
1
VDEm
i i
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Estimation Bound on Anchor-free Localization
The number of the nodes doesn’t matter
The shape of the sensor network affects the total estimation bound.
– Nodes are uniformly distributed in a rectangular region (R=L1/L2)
– All inter-node distances are measured
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To Anchor or not to Anchor To give absolute positions to the nodes is more challenging.
Bad geometry of anchors results in bad anchored-localization.
– 195.20 vs 4.26
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Outline
Introduction– Range-based Localization as an Estimation Problem
– Cramer-Rao Bounds (CRB)
Estimation Bounds on Localization– Properties of CRB on range-based localization
– Anchored Localization (3 or more nodes with known positions) Lower Bounds Upper Bounds
– Anchor-free Localization
– Different Propagation Models
Conclusions
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So far, have assumed that the noise variance is constant σ2.
Physically, the power of the signal can decay as 1/da
Consequences:
– Rotation and Translation still does not change the Cramer-Rao bounds V(xi)+V(yi)
– J(cθ) =J(θ)/ca, so the Cramer-Rao bound on the estimation of a single node : ca V(xi), ca V(yi).
Received power per node:
Cramer-Rao Bounds on Localization in Different Propagation Models
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PR converges for a>2 , diverges for a≤2.
Consistent with the CRB (anchor-free).
Cramer-Rao Bounds on Localization in Different Propagation Models
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Outline
Introduction– Range-based Localization as an Estimation Problem
– Cramer-Rao Bounds (CRB)
Estimation Bounds on Localization– Properties of CRB on range-based localization
– Anchored Localization (3 or more nodes with known positions) Lower Bounds Upper Bounds
– Anchor-free Localization
– Different Propagation Models
Conclusions
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Conclusions
Implications on sensor network design:– Bad local geometry leads to poor localization performance.
– Estimation bounds can be lower-bounded using only local geometry.
Implications on localization scheme design: – Distributed localization might do as well as centralized
localization.
– Using local information, the estimation bounds are close to CRB.
Localization performance per-node depends roughly on the received signal power at that node.
It’s possible to compute bounds locally.
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Some open questions
Noise model
– Correlated ranging noises (interference)
– Non-Gaussian ranging noises
Achievability
Bottleneck of localization
– Sensitivity to a particular measurement
– Energy allocation