Andreas Savvides andreas.savvides@yale Office: AKW 212 Tel 432-1275 Course Website

1. Introduction to Location Discovery Lecture 6 September 20, 2005EENG 460a / CPSC 436 / ENAS 960 Networked Embedded Systems &Sensor Networks Andreas Savvides andreas.savvides@yale.edu Office: AKW 212 Tel 432-1275 Course Website http://www.eng.yale.edu/enalab/courses/2005f/eeng460a

2. Lecture Outline • Recap from last lecture, Kalman Filters & time update phase • The SCAAT Kalman Filter • Programming project overview • Computing locations using multidimensional scaling (MDS)

3. Kalman Filter Equations From Greg Welch http://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf

4. KF, EKF & SCAAT “A Kalman Filter(KF) is a recursive mathematical procedure for least-squares estimation of a linear system” • Follows a predictor-corrector model • An Extended Kalman Filter is a KF variant of non-linear systems • A SCAAT Kalman Filter operates on a single measurement at a time.

5. SCAAT Kalman Filter Proposed by Greg Welsh and Gary Bishop at UNC in 1997 Used in Hi-Ball Optoelectronic Tracker System [Slide created from Welsh & Bishop, “SCAAT: Incremental Tracking with Incomplete Information”, Sigraph 97]

6. SCAAT Kalman Filter Single-constraint-at-a-time(SCAAT) • Process one measurement at a time C- # of independent constraints for a unique solution N – # of constraints used to derive an estimate Nind - # of independent constraints [Slide from Welsh & Bishop, “SCAAT: Incremental Tracking with Incomplete Information”, Sigraph 97]

7. SCAAT Aims at Temporal Improvements • Computation is faster • KF matrices H, x and z are smaller • The state prediction model described by matrix A needs to change Example system that uses all measurements at once 

8. Programming Project • Implement a SCAAT filter in SOS • Need to worry about • Correctness of implementation • Layering & modules (See project handout)

9. Back to Distance Based Localization… What if the distances between nodes are know? Can we compute a coordinate system without beacons (or anchors)? Yes, one possible way is to use MDS…

10. Obtaining a Coordinate System from Distance Measurements: Introduction to MDS • MDS maps objects from a high-dimensional space to a • low-dimensional space, • while preserving distances between objects. • similarity between objectscoordinates of points • Classical metric MDS: • The simplest MDS: the proximities are treated as distances in an Euclidean space • Optimality: LSE sense. Exact reconstruction if the proximity data are from an Euclidean space • Efficiency: singular value decomposition, O(n3)

11. LOCALIZATION USING MDS-MAP (Shang, et al., Mobihoc’03) • The basic MDS-MAP algorithm: • Given connectivity or local distance measurement, compute shortest paths between all pairs of nodes. • Apply multidimentional scaling (MDS) to construct a relative map containing the positions of nodes in a local coordinate system. • Given sufficient anchors (nodes with known positions), e.g, 3 for 2-D or 4 for 3-D networks, transform the relative map and determine the absolute the positions of the nodes. • It works for any n-dimensional networks, e.g., 2-D or 3-D.

12. Applying Classical MDS • Create a proximity matrix of distances D • Convert into a double-centered matrix B • Take the Singular Value Decomposition of B • Compute the coordinate matrix X (2D coordinates will be in the first 2 columns) NxN matrix of 1s NxN identity matrix NxN matrix of 1s

13. Example: Localization Using Multidimensional Scaling (MDS) (Yi Shang et. al) • The basic MDS-MAP algorithm: • Compute shortest paths between all pairs of nodes. • Apply classical MDS and use its result to construct a relative map. • Given sufficient anchor nodes, transform the relative map to an absolute map.

14. MDS-MAP ALGORITHM • Compute all-pair shortest paths. O(n3) • Assigning values to the edges in the connectivity graph: • Known connectivity only: all edges have value 1 (or R/2) • Known neighbor distances: the edges have the distance values • Apply classical MDS and use its result to construct a 2-D (or 3-D) relative map. O(n3) • Given sufficient anchor nodes, convert the relative map to an absolute map via a linear transformation. O(n+m3) • Compute the LSE transformation based on the positions of anchors. • O(m3),m is the number of anchors • Apply the transformation to the other unknown nodes. O(n)

15. MDS-MAP (P) – The Distributed Version • Set-up the range for local maps Rlm (# of hops to consider in a map) • Compute maps of individual nodes • Compute shortest paths between all pairs of nodes • Apply MDS • Least-squares refinement • Patch the maps together • Randomly pick a node and build a local map, then merge the neighbors and continue until the whole network is completed • If sufficient anchor nodes are present, transform the relative map to an absolute map • MDS-MAP(P,R) – Same as MDS-MAP(P) followed by a refinement phase

16. MDS-MAP(P) (Shang and Ruml, Infocom’04) • The basic MDS-MAP works well on regularly shaped networks, but not on irregularly shaped networks. • MDS-MAP(P) (or MDS-MAP based on patches of local maps) • For each node, compute a local relative map using MDS • Merge/align local maps to form a big relative map • Refine the relative map based on the relative positions (optional). (When used, referred to as MDS-MAP(P,R) ) • Given sufficient anchors, compute absolute positions • Refinethe positions of individual nodes based on the absolution positions (optional)

17. SOME IMPLEMENTATION DETAILS OF MDS-MAP(P) • For each node, compute a local relative map using MDS • Size of local maps: fixed or adaptive • Merge/align local maps to form a big relative map • Sequential or distributed; scaling or not • Refine the relative map based on the relative positions • Least squares minimization: what information to use • Given sufficient anchors, compute absolute positions • Anchor selection; centralized or distributed • Refinethe positions of individual nodes based on the absolution positions • Minimizing squared errors or absolute errors

18. AN EXAMPLE OF C-SHAPE GRID NETWORKS Known 1-hop distances with 5% range error Connectivity information only MDS-MAP(P) without both optional refinement steps.

19. RANDOM UNIFORMPLACEMENT Connectivity information only Known 1-hop distances with 5% range error 200 nodes; 4 random anchors

20. RANDOM C-SHAPEPLACEMENT Connectivity information only Known 1-hop distances with 5% range error 160 nodes; 4 random anchors

21. Where is the challenge? • The results are numerically correct based on the used measurement model • Two main assumptions • Hops are proportional to distance • Error represented as a function of distance • Is that the case in practice?

22. Back to the Physical Layer: RSSI In Node Localization Ultrasound ToA RSSI in football field Max range 3m, accuracy 2cm Max range 20m, accuracy 7m Power levels: 7,10

23. RSSI Characterization Study(D.Lymberopoulos et. al. based on Chipcon 2420 radio)

24. RSSI: The Log Normal Shadowing Model Problem with indoor environments: Path loss exponent varies in different areas Significantly affected by reflections at higher power levels Antenna factor dominates

25. Transmit Power Levels in Chipcon CC2420 Radio supply voltage= 2.5V And Power = I*V = 1mW = 43.5mW

26. Going from Watts to dBm

27. Sources of RSSI Variability • Intrinsic • Radio receiver and transmitter calibration • Extrinsic • Relative antenna orientation for Receiver/Transmitter • Multipath and shadowing effects

28. Indoor Path Loss Measurements Floor measurements in a 24 x 20 lounge – no obstacles Same power level using suboptimal antenna η=3

29. RSSI Measurement Configuration: RSSI vs Distance • Data cannot be fitted to the widely used log-normal shadowing signal propagation model • Reflections • Antenna Orientation

30. Antenna Radiation Pattern Side View Top View Communication range Symmetric Region Antenna orientation independent regions Communication range

31. Interesting Properties of RSSI Moving away from the antenna active region

32. Link Asymmetry in 3D-scenarios % of one-way links

33. RSSI Study Results & Conclusions • The RX/TX radio calibration issues have minimal effect on localization • If you were using the log normal shadowing model, that would result in approximately 0.5m of error • Antenna orientation is a major source of variability • There is a small region around the antenna, that is less susceptible to orientation effects • This allows to say when 2 radios are in close proximity to each other • Interesting radio property • There are some signal strength measurements that can only be obtained if two antennas(& radios) are very close to each other!

34. Understanding Fundamental Behaviors What is the fundamental error behavior? Measurement technology perspective • Acoustic vs. RF ToF (2cm – 1.5m measurement accuracy) • Distances vs. Angules Deployment - what density? Scalability How does error propagate? Beacon density & beacon position uncertainty Intrinsic vs. Extrinsic Error Component

35. Estimated Location Error Decomposition Channel Effects Setup Error Computation Error Induced by intrinsic measurement error Position Error

36. Cramer Rao Bound Analysis • Cramer-Rao Bound Analysis on carefully controlled scenarios • Classical result from statistics that gives a lower bound on the error covariance matrix of an unbiased estimate • Assuming White Gaussian Measurement Error • Related work • N. Patwari et. al, “Relative Location Estimation in Wireless Sensor Networks”

37. Density Effects Results from Cramer-Rao Bound Simulations based on White Gaussian Error Range Tangential Error m/rad RMS Location Error RMS Location Error/sigma m/m Range Error Scaling Factor Density (node/m2) 20mm distance measurement certainty == 0.27 angular certainty

38. Density Effects with Different Ranging Technologies 6 neighbors 12 neighbors RMS Error(m)

39. Network Scalability Error propagation on a hexagon scenario (angle measurement) Rate of error propagation faster with distance measurements but Much smaller magnitude than angles RMS Location Error x 10 y-coordinate(m) x-coordinate(m)

40. More Observations on Network Scalability… • Performance degrades gracefully as the number of unknown nodes increases. • Increasing the number of beacon nodes does not make a significant improvement • Error in beacons results in an overall translation of the network • Error due to geometry is the major component in propagated error