Localization in V2X Communication Networks
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Presentation made by Alireza Ghods and given by Dr. Stefano Severi at CCP Workshop co-located with IEEE Intelligent Vehicles Conference, 19th June 2016 Gothenburg (Sweden)
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Localization in V2X Communication Networks
- 1. Localization in V2X Communication Networks Alireza Ghods, Stefano Severi, Giuseppe Abreu s.severi@jacobs-university.de School of Engineering & Science - Jacobs University Bremen (GERMANY) June 19, 2016
- 2. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results Downtown Chicago Typical Dense Urban Environment CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 2/21
- 3. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results Dense Urban Environment Typical Urban Canopy Corridor CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 3/21
- 4. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results Urban Canopy Corridor Typical Distribution of GPS RSSI CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 4/21
- 5. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results Location Forwarding over a V2V Network CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 5/21
- 6. Typical Dense Urban Environment Cooperative Network Localization Model and Notation 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 the periphery have estimated positions (subject to errors) Anchor location errors described by covariance matrix Σk CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 6/21
- 7. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results Ranging Model For each j-th hop: ˜dj ∼ (dj, σ2 j ) σ2 j σ2 0 · dj d0 α where α ≥ 0 is pathloss factor and σ2 0 is the ranging variance at a reference distance d0. For a complete multihop path: ¯dk nk ˜dj, ¯σ2 k nk σ2 j . where nk is number of hops CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 7/21
- 8. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results Fundamental Error Limit The FIM and the MSE The covariance matrix associated with the location estimate of a single target ˆθ is Ωθ E (ˆθ − θ)(ˆθ − θ)T The Cramér-Rao lower bound (CRLB) relates Ωθ to the Fisher Information Matrix Ωθ F−1 θ Fθ ∝ N( ¯dk, σk) Anchor uncertainty not considered! CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 8/21
- 9. Typical Dense Urban Environment Cooperative Network Localization Model and Notation 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 ukuT k where k is the anchor’s index and the information vector is uk = ∂ ak − θ ∂θ Fk = 1 ¯dk [(xak − xθ), (yak − yθ)]T Fk Fk = 1 ¯σ2 k 1 + α2 σ2 0 2 dα 0 ( ak − θ )α−2 in which Fk is the information intensity CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 9/21
- 10. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results FIM with Anchor Uncertainty Augmented Parameter Vector Augmented parameter vector θ Θ = θT , aT 1, aT 2, · · · , aT K T Hence ΩΘ E ( ˆΘ − Θ)( ˆΘ − Θ)T ΩΘ F−1 Θ CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 10/21
- 11. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results FIM with Anchor Uncertainty Augmented 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 uT k, 01×η·(k−1), −uT k, 01×η·(K−k) T CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 11/21
- 12. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results Decomposing the Augmented FIM The Multihop Component The multi hop component of FΘ becomes FM = K k=1 vkvT k = A BT B C , where A K k=1 ukuT k BT −u1uT 1, · · · , −uKuT K C diag u1uT 1, · · · , uKuT K CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 12/21
- 13. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results Decomposing the Augmented FIM Adding 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 , CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 13/21
- 14. Typical Dense Urban Environment Cooperative Network Localization Model and Notation 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 ukuT k − K k=1 ukuT k Σ−1 k + ukuT k −1 ukuT k, = K k=1 uk 1 − uT k Σ−1 k + ukuT k −1 uk uT k, = K k=1 uk 1 − uT k Σk − ΣkukuT kΣk 1 + uT kΣkuk uk uT k, = K k=1 uk 1 − uT kΣkuk + uT kΣkukuT kΣkuk 1 + uT kΣkuk uT k, where we used the Sherman-Morrison formula CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 14/21
- 15. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results Relevant Minor: Schur Complement Simplifying further... F∗ θ = K k=1 uk 1 − uT kΣkuk + uT kΣkukuT kΣkuk 1 + uT kΣkuk uT k, = K k=1 uk 1 − νk + ν2 k 1 + νk uT k, = K k=1 1 1 + νk ukuT k, where νk uT kΣkuk Anchor uncertainty appears as a reduction of information intensity CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 15/21
- 16. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results Some Results ... One-dimensional and two-dimensional scenarios considered 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 the network CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 16/21
- 17. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results 0 50 100 150 200 250 300 350 400 450 500 0 0.5 1 1.5 2 2.5 3 3.5 Monodimensional Scenario Performance for different GPS errors, SNR = 5dB ErrorStandarDeviationε Road Length [m] GPS Σ = 0.9 GPS Σ = 0.5 No GPS Error Anchor Vehicles Selected Targets CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 17/21
- 18. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results 0 50 100 150 200 250 300 350 400 450 500 0 1 2 3 4 5 6 Monodimensional Scenario Performance for different SNR ErrorStandarDeviationε Road Length [m] SNR = 0 dB SNR = 5 dB SNR = 10 dB Anchor Vehicles Selected Targets CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 18/21
- 19. Typical Dense Urban Environment Cooperative Network Localization Model and Notation Ranging Model FIM Formulation Anchor Uncertainty Results 0 50 100 150 200 250 300 350 400 450 500 0 1 2 3 4 5 6 7 8 9 10 Bidimensional Scenario Error Bounds on x-Dimension for Selected Targets with SNR = 5 dB RoadWidth[m] Road Length [m] Anchors Vehicles Target Vehicles CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 19/21
- 20. Typical Dense Urban Environment Cooperative Network Localization Model and Notation 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 σ2 y From that, error ellipsis with diameters λx 1 2 σ2 x + σ2 y − (σ2 x − σ2 y )2 + 4σ2 xy λy 1 2 σ2 x + σ2 y + (σ2 x − σ2 y )2 + 4σ2 xy A numerical example: θ = 464.0172 7.1399 Xa = 0 500.0000 2.5000 2.5000 F = 1.1425 0.0010 0.0010 0.0021 Ω = 0.8757 −0.4230 −0.4230 479.9479 CCP Workshop 2016 Localization in V2X Communication Networks June 19, 2016 20/21
- 21. Thank you!