Localization in V2X Communication Networks

Localization in V2X CommunicationNetworks

Alireza Ghods, Stefano Severi, Giuseppe [email protected]

School of Engineering & Science - Jacobs University Bremen (GERMANY)

June 19, 2016

Typical DenseUrbanEnvironment

CooperativeNetworkLocalizationModel and Notation

Ranging Model

FIM Formulation

Anchor Uncertainty

Results

Downtown ChicagoTypical Dense Urban Environment

CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 2/21



Ranging Model

FIM Formulation

Anchor Uncertainty

Results

Dense Urban EnvironmentTypical Urban Canopy Corridor




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

Urban Canopy CorridorTypical Distribution of GPS RSSI




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

Location Forwarding over a V2V Network




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

Model and Some Notation

A network of with N vehicles in η-dimensional space

[θ1, . . . ,θnT ,anT+1 , . . . ,aN ]

dij , ‖θi − θj‖ =√〈θi − θj ,θi − θj〉

First nT vehicles (targets) have unknown positions

K = N − nT of the remaining vehicles (anchors) in theperiphery have estimated positions (subject to errors)

Anchor location errors described by covariance matrix Σk




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

Ranging Model

For each j-th hop:

dj ∼ (dj , σ2j )

σ2j , σ2

0 ·(djd0

)αwhere α ≥ 0 is pathloss factor and σ2

0 is the rangingvariance at a reference distance d0.For a complete multihop path:

dk ,nk∑

dj ,

σ2k ,

nk∑σ2j .

where nk is number of hopsCCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 7/21



Ranging Model

FIM Formulation

Anchor Uncertainty

Results

Fundamental Error LimitThe FIM and the MSE

The covariance matrix associated with the locationestimate of a single target θ is

Ωθ , E[(θ − θ)(θ − θ)T

]The Cramér-Rao lower bound (CRLB) relates Ωθ to theFisher Information Matrix

Ωθ F−1θ

Fθ ∝ N (dk, σk)

Anchor uncertainty not considered!




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

Constructing the FIM

Standard: element-wise derivative of log-likelihood function

Alternative: sum of products of information vectors

Fθ =∑k∈K

ukuTk

where k is the anchor’s index and the information vectoris

uk =∂‖ak − θ‖

∂θ

√Fk =

1

dk[(xak − xθ), (yak − yθ)]

T√Fk

Fk =1

σ2k

(1 +

α2 σ20

2 dα0(‖ak − θ‖)α−2

)in which Fk is the information intensity




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

FIM with Anchor UncertaintyAugmented Parameter Vector

Augmented parameter vector θ

Θ =[θT, aT

1, aT2, · · · , aT

K

]THence

ΩΘ , E[(Θ−Θ)(Θ−Θ)T

]

ΩΘ F−1Θ




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

FIM with Anchor UncertaintyAugmented Information Vectors

The FIM of Θ can be approximated by (Bayesian rule)

FΘ ≈ FM + FΣ,

where FM accounts for the multi hop ranging, while FΣ

accounts for anchor uncertainty

The approximation holds whenever ‖θ − ak‖ tr(Σk), ∀ k

The extended information vector is then

vk ,∂‖ak − θ‖

∂Θ=

1√Fk

[uTk, 01×η·(k−1), −uT

k, 01×η·(K−k)

]T




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

Decomposing the Augmented FIMThe Multihop Component

The multi hop component of FΘ becomes

FM =

K∑k=1

vkvTk =

[A BT

B C

],

where

A ,K∑k=1

ukuTk

BT ,[−u1u

T1, · · · , −uKuT

K

]C , diag

(u1u

T1, · · · , uKuT

K

)




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

Decomposing the Augmented FIMAdding the Anchor Uncertainty Component

The anchor uncertainty component FΘ is

FΣ ,

[0η×η 0η×ηK

0Kη×η Σ−1

]where Σ , diag (Σ1, · · · , ΣK).

Finally

FΘ ≈[

A BT

B C + Σ−1

],




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

Relevant Minor: Schur Complement

Taking η × η Schur complement of FΘ

F∗θ = A−BT (Σ−1 + C)−1

B,

=

K∑k=1

ukuTk −

K∑k=1

ukuTk

(Σ−1k + uku

Tk

)−1uku

Tk,

=

K∑k=1

uk

(1− uT

k

(Σ−1k + uku

Tk

)−1uk

)uTk,

=

K∑k=1

uk

(1− uT

k

(Σk −

ΣkukuTkΣk

1 + uTkΣkuk

)uk

)uTk,

=

K∑k=1

uk

(1− uT

kΣkuk +uTkΣkuku

TkΣkuk

1 + uTkΣkuk

)uTk,

where we used the Sherman-Morrison formulaCCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 14/21



Ranging Model

FIM Formulation

Anchor Uncertainty

Results

Relevant Minor: Schur Complement

Simplifying further...

F∗θ =

K∑k=1

uk

(1− uT

kΣkuk +uTkΣkuku

TkΣkuk

1 + uTkΣkuk

)uTk,

=

K∑k=1

uk

(1− νk +

ν2k

1 + νk

)uTk,

=

K∑k=1

1

1 + νkuku

Tk,

where νk , uTkΣkuk

Anchor uncertainty appears as areduction of information intensity




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

Some Results ...

One-dimensional and two-dimensional scenariosconsidered

Road: 500 meters long, 10 wide

Only vehicles at borders can self-localize via GPS

Neighborhood set: dij ≤ 70 meters

How well GPS location estimates propagate through thenetwork




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

0 50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

2.5

3

3.5

Monodimensional ScenarioPerformance for different GPS errors, SNR = 5dB

Error

Standar

Deviation

ε

Road Length [m]

GPS Σ = 0.9GPS Σ = 0.5No GPS Error

Anchor Vehicles

Selected Targets




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

Monodimensional Scenario

Performance for different SNR

Error

Standar

Deviation

ε

Road Length [m]

SNR = 0 dBSNR = 5 dBSNR = 10 dB

Anchor Vehicles

Selected Targets




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

7

8

9

10

Bidimensional ScenarioError Bounds on x-Dimension for Selected Targets with SNR = 5 dB

RoadW

idth

[m]

Road Length [m]

Anchors Vehicles

Target Vehicles




Ranging Model

FIM Formulation

Anchor Uncertainty

Results

Why the Huge Errors in Y-Axis

In 2D the covariance matrix is

Ω∗θ =

[σ2

x σxy

σxy σ2y

]From that, error ellipsis with diameters

λx ,1

2

[σ2

x + σ2y−√

(σ2x − σ2

y )2 + 4σ2xy

]λy ,

1

2

[σ2

x + σ2y+√

(σ2x − σ2

y )2 + 4σ2xy

]A numerical example:

θ =

(464.01727.1399

)Xa =

(0 500.0000

2.5000 2.5000

)

F =

(1.1425 0.00100.0010 0.0021

)Ω =

(0.8757 −0.4230−0.4230 479.9479

)CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 20/21

Thank you!