1. Introduction to Location Discovery Lecture 4 September 14, EENG 460a / CPSC 436 / ENAS 960 Networked Embedded Systems & Sensor Networks
Andreas Savvides
Office: AKW 212
Tel
Course Website

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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
Multipath, shadowing, sensor imperfections, changes in propagation properties and more
Extensive computation aspects
Many formulations of localization problems, how do you solve the optimization problem?
How 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
Nodes have to collaborate and communicate to solve the problem
If 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
How do you build a whole system for localization?
How do you integrate location services with other applications?
Different 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 sound1 2 3 4
Minimize over all
This can be linearized to the form
where
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
Unknown node
(known position)
Beacon node
(known position)

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

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 is1 2 3 4 5 6
The objective function is
Can be solved using iterative least squares utilizing the initial
Estimates from phase 2 - we use a Kalman Filter

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 U a a a x

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 Y
b+c
b+c c b U a X
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 Y
b+c
b+c c b U a a a X

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
Challenges
Computation constraints
Communication cost 1 2 3 4 5 2 1 3 4 5 1 2 3 4 5

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

23. Satisfy Global Constraints with Local Computation
From SensorSim
simulation
40 nodes, 4 beacons
IEEE 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
# of edges
# of unknown nodes x 2
Matrices grow with density and number of nodes => so does computation cost
Computation is not feasible on small processors with limited computation and memory

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!10
CR-Bound Evaluation on a 10 x 10 grid
(x,y)
(0,0) 10
beacon
unknown
RMS Error(m)

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

29. Does this solve the problem?
No! Several other challenges
Solution depends on
Problem setup
Infrastructure assisted (beacons), fully ad-hoc & beaconless, hybrid
Measurement technology
Distances vs. angles, acoustic vs. rf, connectivity based, proximity based
The 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…