Introduction to Location Discovery Lecture 9 October 5, 2006 EENG 460a / CPSC 436 / ENAS 960 Networked Embedded Systems & Sensor Networks Andreas Savvides Office: AKW 212 Tel Course Website

Lecture Outline  Programming assignment due next Tuesday You should be making good progress with it!!!  Recap from last lecture  Look at the multihop case  Discuss an EKF solver  Some words on bounds

Last Lecture Recap: 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

Two Possible Ways to Solve the Problem 1.Linearize by subtracting one equation from the rest  Solve the resulting set of linear equations 2. Linearize using Taylor Expansion

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

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

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

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

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

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

Problem Formulation The objective function is Can be solved using iterative least squares utilizing the initial Estimates from phase 2 - we use an Extended Kalman Filter

Things to look out for…  Iterative methods and gradient decent methods are not guaranteed to converge to the global minimum, it is very easy to get stack in local minima To converge properly, you need to provide a good set of initial estimates.  In general this is not a problem with the 1- hop setups but it becomes a big problem with multihop setups

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]

Computation Options 1.Centralized Only one node computes 2. Locally Centralized Some of unknown nodes compute 3. (Fully) Distributed Every unknown node computes Computing Nodes Each approach may be appropriate for a different application Centralized approaches require routing and leader election Fully distributed approach does not have this requirement

Find nodes with unique position solutions Compute Initial Position Estimates For all nodes Compute location estimates Compute estimate at each node Communicate Criteria met? YES NO Communicate results to central point Transmit estimates back to each unknown node Refine estimates of under-constrained nodes Done Centralized ComputationDistributed Computation PHASE 1 PHASE 2 PHASE 3: Refinement

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

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]

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

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

Overview: Collaborative Multilateration Collaborative Multilateration Challenges Computation constraints Communication cost

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

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

Kalman Filter Equations 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

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

Kalman Filter Equations From Greg Welch

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.

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]

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 N ind - # of independent constraints [Slide from Welsh & Bishop, “SCAAT: Incremental Tracking with Incomplete Information”, Sigraph 97]

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 

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]

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…

Fundamental Behaviors and Trends 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

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

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”

Density Effects Density (node/m 2 ) RMS Location Error 20mm distance measurement certainty == 0.27 angular certainty Range Error Scaling Factor RMS Location Error/sigma Range Tangential Error Results from Cramer-Rao Bound Simulations based on White Gaussian Error m/rad m/m

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

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

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