0 IEEE SECON 2004 Estimation Bounds for Localization October 7 th, 2004 Cheng Chang EECS Dept,UC Berkeley Joint work with Prof. Anant Sahai (part of BWRC-UWB project funded by the NSF)

1 IEEE SECON 2004 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

2 IEEE SECON 2004 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

3 IEEE SECON 2004 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

4 IEEE SECON 2004 Localization is an estimation problem Knowledge of anchor positions Range observations: –adj(i) = set of all neighbor nodes of node I –d i,j : distance measurement between node i and j –d i,j = d i,j true + n i,j ( n i,j is modeled as iid Gaussian throughout most of the talk) Parameters to be estimated :

5 IEEE SECON 2004 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={(x i,y i ) T | i є S} is a parameter vector, θ*={R(x i,y i ) T +(a,b) | i ∊ S} is an equivalent parameter vector, where RR T =I 2. Performance evaluation –Anchored: Squared error for individual nodes –Anchor-free: Total squared error

6 IEEE SECON 2004 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 :

7 IEEE SECON 2004 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

8 IEEE SECON 2004 FIM for localization (a geometric interpretation) FIM of the Localization Problem (anchored and anchor-free) –n i,j are modeled as iid Gaussian ~N(0, σ 2 ). –Let θ=(x 1,y 1,…x m,y m ) be the parameter vector, 2m parameters.

9 IEEE SECON 2004 The standard Cramer-Rao bound analysis works. (FIM nonsingular in general) –V(x i )=J( θ) -1 2i-1,2i-1 and V(y i ) =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 )=(x i,y i )+(T x,T y ) –V(x i )+V(y i )=V(x* i )+V(y* i ), if (x* i,y* i )=(x i,y i )R, where RR T =I 2 –J( θ)= J(a θ), if a ≠ 0 Properties of CRB for Anchored Localization

10 IEEE SECON 2004 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

11 IEEE SECON 2004 Local lower bounds: how good can you do? A lower bound on the Cramer-Rao bound, write θ l =(x l,y l ) J l is a 2×2 sub-matrix of J(θ). Then for any unbiased estimator, E(( -θ l ) T ( -θ l ))≥ J l -1 J l only depends on (x l,y l ) and (x i,y i ), i ∊ adj(l) so we can give a performance bound on the estimation of (x l,y l ) using only the geometries of sensor l 's neighbors. Sensor l has W neighbors ( W=|adj(l)| ), then

12 IEEE SECON 2004 Lower bound: how good can you do? J l is the FIM of another estimation problem of (x l,y l ): knowing the positions of all neighbors and inter-node distance measurements between node l and its neighbors.

13 IEEE SECON 2004 Lower bound: how good can you do? J l is the FIM of another estimation problem of (x l,y l ): knowing the positions of all neighbors and inter-node distance measurements between node l and its neighbors.

14 IEEE SECON 2004 Lower bound: how good can you do? J l 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)

15 IEEE SECON 2004 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.

16 IEEE SECON 2004 Upper Bound: what ’ s the best you can do with local information.

17 IEEE SECON 2004 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

18 IEEE SECON 2004 Equivalent class in the anchor-free localization If α={(x i,y i ) T | i є S}, β={R(x i,y i ) T +(a,b) | i ∊ S} is equivalent to α, where RR T =I 2. –Same inter-node disances A parameter vector θ α is an equivalent class θ α ={β |β={R(x i,y i ) T +(a,b) | i ∊ S} }

19 IEEE SECON 2004 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 λ i, 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

20 IEEE SECON 2004 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=L 1 /L 2 ) –All inter-node distances are measured

21 IEEE SECON 2004 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. – vs 4.26

22 IEEE SECON 2004 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

23 IEEE SECON 2004 So far, have assumed that the noise variance is constant σ 2. Physically, the power of the signal can decay as 1/d a Consequences: –Rotation and Translation still does not change the Cramer-Rao bounds V(x i )+V(y i ) –J(c θ) = J( θ)/c a, so the Cramer-Rao bound on the estimation of a single node : c a V(x i ), c a V(y i ). Received power per node: Cramer-Rao Bounds on Localization in Different Propagation Models

24 IEEE SECON 2004 P R converges for a>2, diverges for a≤2. Consistent with the CRB (anchor-free). Cramer-Rao Bounds on Localization in Different Propagation Models

25 IEEE SECON 2004 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

26 IEEE SECON 2004 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.

27 IEEE SECON 2004 Some open questions Noise model –Correlated ranging noises (interference) –Non-Gaussian ranging noises Achievability Bottleneck of localization –Sensitivity to a particular measurement –Energy allocation