Wireless Localization: Positioning

Wireless Localization: Positioning

Stefano Severi and Giuseppe [email protected]

School of Engineering & Science - Jacobs University Bremen

October 7, 2015

Follow-Up Lect Iand II

PositioningPreliminaries

LocalizationAlgorithmsPreliminaries

Least Squares

MDS

Smacof

Cramèr-Raolower boundTarget Specific Approach

Bidimensional Scenario

ConfidenceDegree

Open QuestionFacing the Ranging Error

What happens when all measurements are error-affected?

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Positive Ranging Error

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AnchorsTarget

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Error on RangingForced to Face it!

Ranging Error

We have observed that Ranging Error, even in perfectLine-of-Sight conditions, can not be avoided and neither canbe negligible.





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PositioningLocalization over Geographic Support

Localization Algorithm

The goal is to obtain a quite accurate estimation of a targetposition even in presence of ranging error.





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PositioningNetworks

Different type of Localization Networks

Multihop Network Fully Connected Network





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PositioningDifferent type of Localization Scenarios

C(Θ) is the convex hull of ΘA, i.e. the smallest convex setthat contains the anchor nodes.

Target inside C(Θ) Target outside C(Θ)





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Preliminaries Definitions

θa1 , · · ·,θaN is the ordered coordinate vectors of the aN

known nodes (a-priori known location).

θ1, · · ·,θnT is the ordered coordinate vectors of the nT

target nodes (unknown location).

ΦA = [φa1 , · · · , φηaN ]T and Φ = [φ1, · · · , φηnT ]T are thestacked vectors whose first and second nT elements aregiven by θi:x and θi:y, respectively, where T denotestranspose.

dij , ‖θi − θj‖ =√〈θi − θj ,θi − θj〉 is the true distance

between the i-th and j-th nodes.

eij ∼ N (0, σ2ij) is the Gaussian ranging error on the

distance estimate between the i-th and j-th nodes.

dij = dij + eij is the measured distance





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Least SquaresEffect of Ranging Error

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Least-Square Example without Ranging Error

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LS without ranging error

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Least-Square Example with Ranging Error

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LS with ranging error





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Least SquaresExploiting the Cost Function

Θ , arg min∑i∈ΘA

∑j∈ΘT

(dij − dij)2

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LS without ranging error(3D error function)

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LS with ranging error(3D error function)





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Least SquaresExploiting the Cost Function

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Least-Square Example without Ranging ErrorY

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LS without ranging error(error function)

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Least-Square Example with Ranging Error

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LS with ranging error(error function)





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Least SquaresDerivation

Problem definition:


∑j∈ΘT

(dij − dij)2, (1)

or, equivalently:


∑j∈ΘT

(‖θi − θj‖ − dij)2. (2)

We have now a non-linear system, often quite complicated tobe solved.





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Linear Least SquaresLinearization

Non-linear Least Squares algorithms use mathematical solutionsto find the parameters (i.e. ΘT ) that minimize the cost function.They typically need an initial guess to start the iterative process(successive refinement of the estimated parameters) and in such aclass of problem a closed-form solution does not exist.

A possible alternative is given by the Linear Least Squares.

The Linearization IdeaWe look for an algebraic manipulation that allows to transform thenon-linear system into a linear system, therefore much simpler tobe treated and solved (and later implemented) using a veryefficient matrix form.





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Linear Least SquaresSingle-Target Linearization Example

When considering one single target per time at θ, solving (2)is equivalent to solve the following system:

‖θi − θ‖ = di, i = 1, . . . , n (3)

and, in turn, to:

(θi:x − θx)2 + (θi:y − θy)

2 = d2i , i = 1, . . . , n. (4)

describing a system of n equations that can be linearizedpivoting one arbitrary equation as follows:

(θi:x−θx)2− (θn:x−θx)

2 +(θi:y−θy)2− (θn:y−θy)

2 = d2i − d2

n, (5)

with i = 1, . . . , (n− 1).





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Linear Least SquaresSingle Target Linearization Example

Expanding and properly grouping all the terms, we obtain:

2θx(θn:x−θi:x)+2θy(θn:y−θi:y) = θ2n:x−θ2

i:x+θ2n:y−θ2

i:y+d2i−d2

n, (6)

with i = 1, . . . , (n− 1),

that is an overdetermined linear system that we can describe inmatrix form as follows:

A θ = b. (7)





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Linear Least SquaresSingle Target Linearization Example

The matrix of eq. (7) are related to the elements of eq. (6) asfollows:

A = 2

(θn:x − θ1:x) (θn:y − θ1:y)(θn:x − θ2:x) (θn:y − θ2:y)

. . . . . .(θn:x − θ(n−1):x) (θn:y − θ(n−1):y)

, (8)

and

b =

θ2θ2n:x−1:x + θ2

n:y − θ21:y + d2

1 − d2n

θ2θ2n:x−2:x + θ2

n:y − θ22:y + d2

2 − d2n

. . .

θ2n:x − θ2

(n−1):x + θ2n:y − θ2

(n−1):y + d2(n−1) − d

2n

. (9)





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Linear Least SquaresMultiple Target Nodes Linearization

Eq. (7) can be generalized for multiple target nodes:

A ΘT = b. (10)

In general A is not invertible, therefore we need to solve theequivalent normal equation linear system:

ATA ΘT = ATb, (11)

then, using the psesudoinverse of ATA, we get to the finalformulation:

ΘT = (ATA)−1AT︸︷︷︸Moore-Penrosepseudoinverse

b. (12)





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Linear Least SquaresMatlab Implementation

A Smart Algebraic Implementation

The backslash operator A\b is used by matlab to solve linearsystems and in the considered case is equivalent to the explicitformulation pinv(A)*b.





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MultiDimensional ScalingThe Fundamental Idea

How MultiDimensional Scaling works:

The distance matrix D can be seen as a dissimilaritiesmatrix between nodes.

Exploiting this information it is possible to map the nodesinto an η-dimensional space.

This mapping is unique but subject to rotation.

It is possible to exploit the information on the a-priori knownnodes (anchors) to get back to the true network realization.





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MultiDimensional ScalingClassic Formulation

Define the Gramian Matrix

KΘ , ΘTΘ. (13)

It can be shown being a rotation of the positive semidefiniteand double-centered

KD , −1

2J ·D2 · J, (14)

where m denotes the m-th element-wise power, the matrixJ is given by

J , I− 1

N(1 · 1T), (15)

I is identity matrix and 1 is a vector whose entries are all 1.





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Question: Why we want to use KD rather than KΘ?

Answer: Because we do not know Θ (is the goal of ourlocalization process) but we have some knowledge aboutD. We have infact measured D.

We can now decompose KD as follows:

KD , UD ·ΛD ·UTD, (16)

where UD and ΛD are, respectively, the eigenvector andeigenvalue matrices of KD.





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A rotated and translated matrix containing all the nodeslocation can now be obtained from equation (17):

ΘR =

(UD

N×η·ΛD

12

η×η

)T

, (17)

where · m×n denotes the m-by-n upper-left partition andthe symbol R denotes the rotation.

Since at least η + 1 column of Θ are known, it is possible toget an estimated matrix Θ from ΘR via Procrustestransformation.





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Analyzing MDS

The MultiDimensional Scaling:

In the above scenario (network + ranging error) has thesame performances than LS.

Zero error in absence of ranging error.

It is now presented as a possible new theoretical approach.

It is the base for one of the most powerful localizationalgorithm Super MDS.





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SmacofCooperation without confidence box

σ(X(0)) =∑

i<j wij(d(0)ij − dij)

2 is the original stress function

d(m)ij is the estimated distance dij after the m-th iteration

X(m) is the estimate of X = [Φ;ΦA] after the m-th iteration

X(m−1) is the solution at the previous iteration

τ(X(m), X(m−1)) is the majored convex function at the m-th iteration

The majorized convex functionThis algorithm attempts to find the minimum of a non-convex function bymajorizing the initial stress function and then tracking the global minimumof the so-called majorized convex function τ(X(m), X(m-1)) ,which is thensuccessively constructed from the previous solution X(m-1) obtained in theiterative procedure.

The expression of the majorized convex function is:

τ(X(m), X(m−1)) = 1+tr(X(m)TVX(m)

)−2tr

(X(m)TB(X(m−1))X(m−1)

)Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 23/33




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SmacofCooperation without confidence boxes

The entries of the matrix V are given by

vij =

∑i=1i6=j

wij i = j

wij i 6= j

while the i-th element of the matrix B(X(m−1)) is defined as follow:

bij =

∑i=1i6=j

wijdij

d(m−1)ij

i = j

wijdij

d(m−1)ij

i 6= j

Since τ(X(m), X(m−1)) is a quadratic convex function, in order tocompute X(m) is sufficient to solve the following equation:

∂τ(X(m), X(m−1))

∂X(m)= 0





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SmacofCooperation without confidence boxes

After the m-th iteration the estimated locations are given by:

X(m) = V†B(X(m−1))X(m−1)

where V† is the Moore-Penrose Pseudoinverse of V.

The iterative process ends when σ(X(m))− σ(X(m−1)) < ε or acertain iteration limit is reached.

Accuracy plus Confidence

The Smacof algorithm is known for its accuracy and efficiency (it isapplicable also to non-convex functions).It doesn’t provide an estimate of the confidence associated withany position estimate.





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CRLBA Concise and General Bound on the Ranging Variance

A general description of the error model:

σ2ij , σ2

0 ·(dijd0

)αα ≥ 0 is the path-loss factor.

d0 is the reference distance.

σ20 is the ranging variance at d0.

The Generality of the Bound

Lower bounds on the ranging errors obtained from distanceestimates using either narrowband or ultra-wideband radios is

σ ≥ β√SNR

where β is a coefficient depending on the speed of light and on thesignal’s duration, center-frequency and bandwitdh.





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Target Specific ApproachThe Covariance Matrix

The idea: a node cooperates only with the subset of othernodes which knows its own position (anchor nodes).

θi is the estimate of the locationof the i-th target

Associated with each θi there isthe η-by-η covariance matrix:

Ωθi , E[(θi − θi)(θi − θi)T

]The CRLB relates this covariance matrix to the inverse FIMby

Ωθi F−1θi





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Target Specific ApproachThe Fisher Information Matrix

The likelihood function of the Target specific approach isgiven by:

f(θi; di) ,NA∏j=1

1σij√

2πexp

(− (‖θi−θj‖−dij)2

2σ2ij

)The anchor-to-target paths is

dij ,

nij∑k=1

dk

Associated Fisher InformationMatrix: the qp−th element isgiven by:

Fθi , −E[∂2 ln f(θi;di)

∂2θi

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Bidimensional ScenarioRegular Planar Network

FIM under Target-specific Approach

Fθi,

[Fθi:xx Fθi:xy

Fθi:xy Fθi:yy

]

Fθi:xy =∑j∈Ni

(∆θij:x ∆θij:yσ2ij d

2ij

+α2 ∆θij:x ∆θij:y

2d4ij

).





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Error Confidence/EllipseThe Uncertainty Ellipse

The Error Ellipse express the confidence region of a target positionestimate, under a given error probability, for a generic χ2-errordistribution.

For each target the Covariance Matrix is: Ωθi =

σ2x σxy

σxy σ2y

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5% Uncertainty Ellipses in a Regular Planar Network

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σ = 0.01σ = 0.03

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The scale factors sf is givenby theinverse of the CDF of the χ2 at thegiven confidence probability

The length of the axis is given bysf ·

√eigval(Ωθi)

The rotation angle β is given by

β = 12atan

[(1

sfx

)(2σxy

σx2−σ2

y

)]Specialization Lab - Fall 2015 Wireless Localization: Positioning October 7, 2015 30/33




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Report 3/3Positioning

Complete the lab experience writing (one per group) a reportwith:

1 create the map of the network environment using thecoordinates of anchor and target given,

2 run in Matlab LS and MDS, obtain the different targetestimated positions using the distances d provided,

3 estimate the average ranging error and variance of the eachtarget distance link,

4 Obtain the Covariance matrix Ωθ and draw the Error ellipseusing the mean of the target’s position estimates andinverse variance of the target’s distance estimates.





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Report 3/3Positioning

5 Save the code, the maps and the plot the target estimatedposition and the fisher ellipse of both LS and MDS indifferent figures.

6 write a short (max 2 pages) description of this experiment.

Please print and deliver the report within the aforementioned deadline to

[email protected],

[email protected].

Matlab TipTo perform the report, use the command provided in the Matlabtips.m file


Thank you!

Wireless Localization: Positioning

Engineering

Transcript of Wireless Localization: Positioning