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
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
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
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
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
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