1. Introduction to Location Discovery Lecture 4 September 14, 2004 EENG 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/eeng460a

2. Today  Announcement of student presentations  Make sure you get the additions from last set of lecture slides  Make sure your name is on signup sheet for the lab  Stop by Ed Jackson’s office so that he can swipe your ID for the lab  Internal website access Some work-in-progress and copyrighted material will only appear on this site. Note the password. Reference material for this lecture: [Savvides 04a]A. Savvides and M. B. Srivastava, Location Discovery, pp 231 – 238 [Savvides 03] A. Savvides, H. Park and M. B. Srivastava, The n-Hop Multilateration Primitive for Node Localization Problems [Goldenberg04] D. Goldenberg, A. Krishnamurthy, W. Maness, Y. R. Yang, A. Young and A. Savvides, Network Localization in Partially Localizable Networks (slide 22 only)

3. Why is Location Discovery(LD) Important?  Very fundamental component for many other services GPS does not work everywhere Smart Systems – devices need to know where they are Geographic routing & coverage problems People and asset tracking Need spatial reference when monitoring spatial phenomena  We will use the node localization problem as a platform for illustrating basic concepts from the course

4. Why spend so much time on LD?  LD captures multiple aspects of sensor networks: Physical layer imposes measurement challenges oMultipath, shadowing, sensor imperfections, changes in propagation properties and more Extensive computation aspects oMany formulations of localization problems, how do you solve the optimization problem? oHow do you solve the problem in a distributed manner? –You may have to solve the problem on a memory constrained processor… Networking and coordination issues oNodes have to collaborate and communicate to solve the problem oIf you are using it for routing, it means you don’t have routing support to solve the problem! How do you do it? System Integration issues oHow do you build a whole system for localization? oHow do you integrate location services with other applications? oDifferent implementation for each setup, sensor, integration issue

5. Base Case: Atomic Multilateration  Base stations advertise their coordinates & transmit a reference signal  PDA uses the reference signal to estimate distances to each of the base stations  Note: Distance measurements are noisy!

6. Problem Formulation  Need to minimize the sum of squares of the residuals  The objective function is  This a non-linear optimization problem Many ways to solve (e.g a forces formulation, gradient descent methods etc

7. A Solution Suitable for an Embedded Processor  Linearize the measurement equations using Taylor expansion where Now this is in linear form

8. Solve using the Least Square Equation The linearized equations in matrix form become Now we can use the least squares equation to compute a correction to our initial estimate Update the current position estimate Repeat the same process until δ comes very close to 0

9. How do you solve this problem?  Check conditions Beacon nodes must not lie on the same line Assuming measurement error follows a white gaussian distribution  Create a system of equations Solve to get the solution – how would you solve this in an embedded system? How do you solve for the speed of sound?

10. Acoustic case: Also solve for the speed of sound Minimize over all This can be linearized to the form where 1 2 3 4 0 MMSE Solution:

11. The Node Localization Problem Beacon nodes Localize nodes in an ad-hoc multihop network Based on a set of inter-node distance measurements

12. Solving over multiple hops  Iterative Multilateration Beacon node (known position) Unknown node (known position)

13. Iterative Multilateration Problems Error accumulation May get stuck!!! % of initial beacons Localized nodes total nodes

14. Collaborative Mutlilateration All available measurements are used as constraints Solve for the positions of multiple unknowns simultaneously Catch: This is a non-linear optimization problem! How do we solve this? Known position Uknown position

15. Problem Formulation The objective function is Can be solved using iterative least squares utilizing the initial Estimates from phase 2 - we use a Kalman Filter 1 2 3 4 5 6

16. How do we solve this problem?  In an embedded system?  Backboard material here…  One possible solution would use a Kalman Filter This was found to work well in practice, can “easily” implemented on an embedded processor [more details see Savvides03]

17. Initial Estimates (Phase 2)  Use the accurate distance measurements to impose constraints in the x and y coordinates – bounding box  Use the distance to a beacon as bounds on the x and y coordinates a aa x U

18. Initial Estimates (Phase 2)  Use the accurate distance measurements to impose constraints in the x and y coordinates – bounding box  Use the distance to a beacon as bounds on the x and y coordinates  Do the same for beacons that are multiple hops away  Select the most constraining bounds a b c b+c X Y U U is between [Y-(b+c)] and [X+a]

19. Initial Estimates (Phase 2)  Use the accurate distance measurements to impose constraints in the x and y coordinates – bounding box  Use the distance to a beacon as bounds on the x and y coordinates  Do the same for beacons that are multiple hops away  Select the most constraining bounds  Set the center of the bounding box as the initial estimate a aa b c b+c X Y U

20. Initial Estimates (Phase 2)  Example: 4 beacons 16 unknowns  To get good initial estimates, beacons should be placed on the perimeter of the network  Observation: If the unknown nodes are outside the beacon perimeter then initial estimates are on or very close to the convex hull of the beacons

21. Overview: Collaborative Multilateration Collaborative Multilateration Challenges Computation constraints Communication cost 1 2 3 4 5 2 1 3 4 5 1 2 3 4 5

22. Overview: Collaborative Multilateration Collaborative Multilateration Challenges Computation constraints Communication cost Distributed reduces computation cost Even sharing of communication cost

23. Satisfy Global Constraints with Local Computation  From SensorSim simulation  40 nodes, 4 beacons  IEEE 802.11 MAC  10Kbps radio  Average 6 neighbors per node

24. Kalman Filter From Greg Welch We only use measurement update since the nodes are static We know R (ranging noise distribution) Not really using the KF for now, no notion of time

25. Global Kalman Filter  Matrices grow with density and number of nodes => so does computation cost  Computation is not feasible on small processors with limited computation and memory # of edges # of unknown nodes x 2

26. Beware of Geometry Effects! Known as Geometric Dilution of Precision(GDOP) Position accuracy depends on measurement accuracy and geometric conditioning i k j From pseudoinverse equation

27. Geometry: It’s the angles not the distance! RMS Error(m) CR-Bound Evaluation on a 10 x 10 grid (0,0) 10 beacon unknown (x,y)

28. Beware of Solution Uniqueness Requirements  In a 2D scenario a network is uniquely localizable if: 1.It belongs to a subgraph that is redundantly rigid 2.The subgraph is 3-connected 3.It contains at least 3 beacons  More details in future lectures Nodes can be exchanged without violating the measurement constraints!!! [conditions from Goldenberg04]

29. Does this solve the problem?  No! Several other challenges  Solution depends on Problem setup oInfrastructure assisted (beacons), fully ad-hoc & beaconless, hybrid Measurement technology oDistances vs. angles, acoustic vs. rf, connectivity based, proximity based oThe underlying measurement error distribution changes with each technology  The algorithm will also change Fully distributed computation or centralized How big is the network and what networking support do you have to solve the problem? Mobile vs. static scenarios Many other possibilities and many different approaches  More next time…