Convex Position Estimation in
Wireless Sensor Networks
Lance Doherty
Kristofer S. J. Pister
Laurent El Ghaoui
Presented by
Nitya Kalathuru and Dharmen Shah
Thursday, 6 March 2002
Objective of the Paper
To estimate unknown node positions in a sensor
network using connectivity constraints.
Given: position of solid
nodes
Find: a possible solution
for each open node
Subject to: proximity
constraints imposed by
known connections
What is
• Proximity/ Connectivity Constraint
- if one node can communicate with another
- these restrict the feasible set of unknown node positions
- these must form a convex set
•
Convex Set
-
Convex
Non-Convex
-
any two points in the set can be
connected with a line contained in
the set
if the line is not completely
contained within the set, the set is
not convex.
Introduction
Where is the data coming from – location of the nodes?
•
Equip all nodes with GPS to know absolute position - costly
•
Inferring positional information from connection-imposed
proximity constraints – very general
In the method proposed
•
A few nodes have known positions – equipped with GPS or placed
deliberately
•
Feasible solutions are obtained using convex optimization
•
Only planar networks are considered
•
Requires centralized computation
•
Authors use math i.e., linear programming, to solve the problem
Mathematical Formulation
•
•
Linear Program (LP)
cTx
Minimize
Subject to Ax < b
c, A, B, Fn – matrices
x - vector
Semidefinite Program (SDP)
Minimize
cTx
Subject to F (x) = F0 + x1F1 +........+ xnFn < 0
Ax < b
Fi = Fi T
- a generalization of LP
- sufficient to solve all numerical problems in this paper
- first inequality is called the Linear matrix inequality
(LMI)
Convex Constraint Models for RF and
Optical Communication Models
•
Connections as convex constraints
Provided that the network connectivity can be represented as a
set of convex position constraints, the mathematical models can be
used to generate feasible positions for the nodes in the network
•
Radial constraint
– RF communication
- RF TX of the node is configured
to have rotationally symmetric
range
- Two types: fixed radial
variable radial
Convex Constraint Models.. cont’d
•
Angular constraint - optical communication
- laser TX and RX rotate and scan through some angle
- RX rotates and calculates the angle first roughly & then finely
- by observing this angle, an estimate of the relative angle & max.
distance to the TX can be obtained
- in 3-D this results in a cone for the feasible set
Convex Constraint Models.. cont’d
•
Other convex constraints
- quadrant detector scheme
- trapezoid
•
Combining Individual
Constraints
- nodes are constrained by
connections to other nodes
- feasible region becomes smaller
with each added constraint
- this is the mechanism used in
position estimation
Summary of Constraints Types
Simulation and Results
Software


tools
Mat lab using Mosek optimization
Hardware


AMD k-6 400 MHz processors
64 Mb RAM
Simulation Test bed..



Node positions in
10R square region
200 Nodes randomly
places
10 Best-connected
networks chosen
Performance Metrics..



Best estimation of node is in the Intersection
of allowable regions
Performance is calculated as the measure of
mean error
Mean error – provides a feasible set
Method Adopted..


Bounding of the feasible set
Use of a Rectangular region for best
estimation approximation
(1)
(2)
(3)

Centroid gives the best approximation
Constraint Results..
Radial and Angular constraints
Procedure


1.
2.
3.
4.
5.
Selecting Node 1 as known position (m=1)
Solve for remaining n-m positions
Computing the mean error for n-m unknown
positions
Increase m by 1
Redo 2-4 until m=100
Results..

Radial
constraints


Fixed radius and
Variable radius
analysis
Variable radius
method is
superior since it
gives flexibility
of distance
sensing
Analysis contd..



Radial constraints have the convex hull on the
position of the unknown nodes
Best results can be obtained by placing the
nodes on the periphery
Experiment results


Using 4-known nodes at the corners, mean error
reduces from 2.4R to 1.2R (variable radius case)
Additional nodes at the centre of the edges
reduces the mean error from 1.7R to 0.72R
Analysis contd..


Using Rectangular
bounds on Radial
constraints
Figure shows :






1,2,3 are known positions
4 thru 7 are unknown
positions
6 -> 1 and 3 ( two known
positions)
5 -> 1 (one known position)
4 -> 2 (one known position
with R=2)
7 -> 2 (one known
position)
Inferences..



Results are drawn by having
8 known node positions with
variable radius
Hence we see that at least 8
nodes are need to achieve
connection bound of 4R2
Two evidences


Nodes with dist R have
rectangle area less than 4R2
(centre nodes with high
connectivity)
Centroid approximation
reduces the error from
0.72R to 0.64R
Results Contd..


For Angular constraints
Two approaches

Using half angle of uncertainty


Varying the Θ from Π/4 to Π/10 and Π/100
Using distance to outer bound

Varying the cone length such as R,2R,4R,10R
Graphs and Inferences..


Smaller individual constraints lead to better position
estimates as seen from the graph
Larger cone length leads to worst results since it causes
more divergence
Analysis..

On Angular
constraints

Effect of increasing
the node density
decreases the mean
error
Conclusions..






Sensor network positioning can be formulated
as a problem of LP or SDP
Variable radius performs better
Placement on known nodes should be on the
boundary of the region
Rectangular bounding improves estimation
For Angle constraints, decreasing
uncertainties reduces the mean error
Increasing the nodes density (Connectivity)
increases the performance of estimation
methods
Applications and Future..






Tracking through sensor network
Hierarchical solution for large networks
Implementing continuous distributions
Combination of angular and radial
constraints
Erroneous data management
Modeling uncertainty in "known"
positions
The End
Questions ????