Sensor/Actuator Sensor/Actuator Network CalibrationNetwork Calibration

Kamin WhitehouseKamin Whitehouse

Nest Retreat, June 17 2002Nest Retreat, June 17 2002

IntroductionIntroduction

Previous sensor systems:Previous sensor systems: ““multi”-sensor = “5” sensormulti”-sensor = “5” sensor Specialized, high-accuracy devicesSpecialized, high-accuracy devices

Sensor networks:Sensor networks: Scores Scores of assembly-line sensorsof assembly-line sensors Non-adjustable, uncalibrated devicesNon-adjustable, uncalibrated devices

Talk OutlineTalk OutlineCalamari OverviewCalamari Overview

General FrameworkGeneral Framework

Noisy environmentNoisy environment Least squaresLeast squares

Partially-unobservable, noisy environmentPartially-unobservable, noisy environment Joint calibrationJoint calibration

Completely unobservable environmentCompletely unobservable environment Constraint-based calibrationConstraint-based calibration

Calamari OverviewCalamari Overview

Simultaneously send sound and RF signalSimultaneously send sound and RF signal

Time stamp both Time stamp both

SubtractSubtract

Multiply by speed of soundMultiply by speed of sound

Filter the readings (one more multiply)Filter the readings (one more multiply)

Calamari ParameterizationCalamari Parameterization

Bias – startup time for mic/sounder Bias – startup time for mic/sounder oscillationoscillation

Gain – Volume and sensitivity affect PLLGain – Volume and sensitivity affect PLL

Frequency -- |FFrequency -- |FTT-F-FRR| is scaling factor| is scaling factor

Orientation – f(OOrientation – f(OTT,O,ORR)) is scaling factoris scaling factorCalibration Function:Calibration Function:

r*= Br*= BTT + B + BRR + G + GTT*r + G*r + GRR*r + |F*r + |FTT-F-FRR|*r + f(O|*r + f(OTT,O,ORR)*r)*r

No Calibration: 74.6% ErrorNo Calibration: 74.6% Error

General Calibration FrameworkGeneral Calibration FrameworkAll calibration is sensor/actuator pairsAll calibration is sensor/actuator pairsIterative Calibration:Iterative Calibration: use single calibrated use single calibrated node to calibrate all other sensors/actuatorsnode to calibrate all other sensors/actuatorsAll sensor/actuator signals are multi-All sensor/actuator signals are multi-dimensionaldimensional Observed signalsObserved signals Unobserved signalsUnobserved signals

Absolute Calibration:Absolute Calibration: choose standard choose standard absolute coordinate scaleabsolute coordinate scaleRelative Calibration:Relative Calibration: choose single node as choose single node as standard coordinate scalestandard coordinate scale

Calibration FunctionCalibration Function

r – measured readingsr – measured readings

r* – desired readingsr* – desired readings

ß ß – parameters– parameters

r* = f(r, ß) r* = f(r, ß)

General Calibration FrameworkGeneral Calibration Framework

Four classes of calibrationFour classes of calibration Known environmentKnown environment Noisy environment or devicesNoisy environment or devices Partially observable environmentsPartially observable environments Unobservable environmentsUnobservable environments

Known EnvironmentKnown Environment

All signals are knownAll signals are known observed observed unobservedunobserved

Implies use of “perfect” calibrating deviceImplies use of “perfect” calibrating deviceCan be used to calibrate all other devicesCan be used to calibrate all other devicesIf devices are uniform:If devices are uniform:

r* = Ar + Br* = Ar + B

If devices have idiosyncrasies:If devices have idiosyncrasies:r* = Ar* = Aiir + Br + Bii

Noisy EnvironmentNoisy Environment

Some input signals are noisySome input signals are noisy I.e. no “perfect” calibrating deviceI.e. no “perfect” calibrating device

Use multiple readings/calibrating devicesUse multiple readings/calibrating devices Assumes noise due to variations has Gaussian distributionAssumes noise due to variations has Gaussian distribution

If devices are uniform:If devices are uniform:

r* = Ar + Br* = Ar + B

If devices have idiosyncrasies:If devices have idiosyncrasies:

r* = Ar* = Aiir + Br + Bii

Uniform Calibration: 21% ErrorUniform Calibration: 21% Error

Noisy Environment: 16%Noisy Environment: 16%

Partially unobservable Partially unobservable

Solve for transmitter and receiver Solve for transmitter and receiver parameters parameters simultaneouslysimultaneously

Assumes noise due to unobserved signal Assumes noise due to unobserved signal has gaussian distributionhas gaussian distribution

If devices are uniform:If devices are uniform:

r* = Ar* = ATTr + Ar + ARRr + Br + BTT + B + BRR

If devices have idiosyncrasies:If devices have idiosyncrasies:

r* = Ar* = Attr + Ar + Arrr + Br + Btt + B + Brr

Joint Calibration: 10.1%Joint Calibration: 10.1%

Auto CalibrationAuto Calibration

No known input signalsNo known input signals

....!?....!?

Constraint-based CalibrationConstraint-based Calibration

All distances in the network must follow All distances in the network must follow the triangle inequalitythe triangle inequality

Let Let ddijij = = BBTT + B + BRR + G + GTT*r + G*r + GRR*r *r

For all connected nodes For all connected nodes i, j, k:i, j, k:

ddij + ij + ddjjkk - d - dikik >=0 >=0

Consistency-based CalibrationConsistency-based Calibration

All transmitter/receiver pairs are also All transmitter/receiver pairs are also receiver/transmitter pairsreceiver/transmitter pairs

These symmetric edges should be equalThese symmetric edges should be equal

Let Let ddijij = = BBTT + B + BRR + G + GTT*r + G*r + GRR*r *r

For all transmitter/receiver pairs For all transmitter/receiver pairs i, j:i, j:

ddik = ik = ddkiki

Quadratic ProgramQuadratic Program

Let Let ddijij = = BBTT + B + BRR + G + GTT*r + G*r + GRR*r *r

Choose parameters to maximize Choose parameters to maximize consistency while satisfying all constraintsconsistency while satisfying all constraints

A quadratic program arisesA quadratic program arisesMinimize: Minimize: ΣΣikik (d(dik ik –– ddkiki))2 + 2 + ΣΣTT(G(GTT–– 11))2 + 2 + ΣΣRR(G(GRR–– 11))22

Subject to: dSubject to: dij + ij + ddjjkk - d - dikik >=0 ; for all triangle >=0 ; for all triangleijijkk

Unobservable Environment: ??%Unobservable Environment: ??%

Future WorkFuture Work

Non-gaussian variations of the above Non-gaussian variations of the above algorithmsalgorithms

Expectation\maximizationExpectation\maximization

MCMCMCMC

ConclusionsConclusions

New calibration problems with sensor New calibration problems with sensor networksnetworks

We can exploit the network itself to solve We can exploit the network itself to solve the problemthe problem Computation on each sensor/actuatorComputation on each sensor/actuator Networking abilityNetworking ability Distributed processingDistributed processing Feedback controlFeedback control