1、Calibration as Parameter Estimation in Sensor NetworksKamin WhitehouseUC BerkeleyBerkeley, CAkamincs.berkeley.eduDavid CullerUC BerkeleyBerkeley, CAcullercs.berkeley.eduABSTRACTWe describe an ad-hoc localization system for sensor net-works and explain why traditional calibration methods areinadequat
2、e for this system. Building upon previous work,we frame calibration as a parameter estimation problem;we parameterize each device and choose the values of thoseparameters that optimize the overall system performance.This method reduces our average error from 74.6% withoutcalibration to 10.1%. We pro
3、pose ways to expand this tech-nique to a method of autocalibration for localization as wellas to other sensor network applications.Categories and Subject DescriptorsB.4.2 Hardware: Input/Output Devices; B.4.5 Hardware:Reliability, Testing, and Fault Toleranceerror checking,hardware reliability; I.2.
4、6 Computing Methodologies:Artificial Intelligenceparameter learningGeneral TermsMeasurement, Reliability, VerificationKeywordsCalibration, Relative Calibration, Auto Calibration, Sensor,Actuator, Sensor Networks, Parameter Estimation1. INTRODUCTIONSensor networks present new challenges in calibratio
5、n.Many sensors today have an easy, built-in calibration in-terface, such as police trac radars that use a tuning fork19. Others are factory calibrated and built to remain cal-ibrated for long periods of time, such as industrial qualityaccelerometers 17. Sensor systems have also traditionallyused onl
6、y a small number of sensors. In robotics, for exam-ple, even “Multi-sensor” systems consist of only a handfulof sensors 4. Even currently deployed sensor “networks”,such as the SeaKeepers Society Ocean Monitoring System,Permission to make digital or hard copies of all or part of this work forpersona
7、l or classroom use is granted without fee provided that copies arenot made or distributed for prot or commercial advantage and that copiesbear this notice and the full citation on the rst page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specicpermi
8、ssion and/or a fee.WSNA02, September 28, 2002, Atlanta, Georgia, USA.Copyright 2002 ACM 1-58113-589-0/02/0009 .$5.00.are of the macro variety, where each sensor is individuallymanaged 18.As a result of this, most sensor-systems are built for micro-calibration, in which each device is individually tu
9、ned in acarefully controlled environment.Sensor networks, however, demand a new method of cal-ibration. Individual calibration of hundreds or thousandsof devices can be problematic, especially when many smallor low-power devices have no calibration interface. Fur-thermore, devices often need to be c
10、alibrated in partiallyunobservable and dynamic environments, or may even beunobservable themselves. They are also often general pur-pose devices that need to be calibrated dierently for eachapplication.The demand for macro-calibration has become real in ourad-hoc localization system for sensor netwo
11、rks. The deviceswe use are so integrated into the system that we cannoteasily observe them for calibration. Even if they could beobserved, there are too many devices to manage each one in-dividually. They do not have calibration interfaces and evenif they did it would be unclear how to use them to c
12、alibrate,for example, an acoustic device for distance estimation.We explore macro-calibration by framing calibration asa general parameter estimation problem. For each device,we choose calibration parameters that optimize the overallsystem response, instead of the individual device responses.There a
13、re many benefits to this technique. First, it freesus from the need to directly observe and calibrate each andevery device; we only need to observe the overall system re-sponse. Second, it provides a calibration interface to thosedevices that do not already have one. This interface is spe-cific to o
14、ur application instead of specific to the device. Fi-nally, it suggests a method of autocalibrating distance esti-mates without actually knowing the true distances betweenthenodes.WecallThe rest of this paper is organized as follows. Section2 introduces our ad-hoc localization system and its cali-br
15、ation problems. Section 3 formalizes calibration in thetraditional sense and explains why it is inadequate for lo-calization. Section 4 discusses two micro-calibration tech-niques for localization while section 5 describes a method ofmacro-calibration. Sections 6 and 7 evaluate and comparethese thre
16、e methods. Section 8 describes an extension ofthis calibration technique to an autocalibration method forlocalization. Secion 9 suggests ways to extend this work toother sensor network application.59 2. AD-HOC LOCALIZATIONAND THE CALIBRATION PROBLEMLocalization becomes a much more dicult problem int
17、he context of ad-hoc sensor networks. Almost all localiza-tion systems existing today rely heavily on infrastructureto provide known positions and distances. GPS and ActiveBats from AT there is only one response from the sys-tem and it cannot separate the eect of the transmitter fromthe eect of the
18、receiver. We will call this the separationproblem. In the following sections, we discuss three meth-ods to overcome the separation problem: iterative, mean,and joint calibration.4. TOWARDS AVOIDINGTHE SEPARATION PROBLEMThe separation problem is not specific to Calamari, but tothe calibration of all
19、sensor/actuator pairs. We realize theimportance of it when we see that all calibration is betweensensor/actuator pairs. We usually do not worry about itwhen calibrating temperature sensors with room tempera-ture, but we might need to when calibrating, for example,magnetometers with a set of magnets.
20、The first attempt we see to circumvent the separationproblem comes from the ad-hoc localization literature. Spo-tON 10 uses low-power radio transceivers in an RSSI rang-ing system very similar to the RSSI component of Calamari.To calibrate, SpotON observes that the seperation problemcould be avoided
21、 if we had at least one calibrated transmitteror receiver. As a solution, it simply declares one transmitterto be the reference and uses it to calibrate all receivers. Itthen uses one reference receiver to calibrate all transmitters.This procedure eectively iterates traditional calibration,so we wil
22、l call it iterative calibration. In both iterations thereis only one uncalibrated device in each pair. When a mea-surement from the pair is in error, the error is attributed tothe uncalibrated device, thereby getting past the separationproblem.An interesting part of the SpotON technique is that ther
23、eceivers are not actually adjusted at all. Instead, each re-ceiver is parameterized in terms of the Seidel and RappaportRF path loss model. There are two main advantages to usingthis parameterization. First, the receiver sensitivity cannotbe adjusted in hardware anyway. Second, and more im-portantly
24、, the parameterization can be arbitrarily accurate.In fact, Hightower, et al take advantage of gaussian noise intheir data by eectively using least-squares log-linear regres-sion over 100 readings. This technique is expanded upon inthe present work.There is one serious problem with iterative calibra
25、tion asapplied to SpotON. RSSI is eected by the dierence in fre-quency of the transmitter and receiver, as seen in Figure1. This means a receivers parameters are only valid whenthe transmitter it is currently paired with has the same fre-quency as the transmitter used to calibrate it. iterative cal-
26、ibration does not actually avoid the seperation problem atall; calibration parameters are only known to be valid forone transmitter/receiver pair.Let us propose one-more attempt to avoid the separationproblem. Here, we assume that variations in the devices aregaussian distributed. By doing so, we ca
27、n calibrate all re-ceivers to the mean of the transmitters, or vice versa. Letus call this method mean calibration. In mean calibrationone collects calibration data for each receiver using all trans-mitters as calibrating devices. The parameters learned forthe receiver will not be coupled to any par
28、ticular transmitterand will also minimize the expected error for that receiver inthe least-squares sense, assuming that the sample of trans-mitters used represents the true transmitter population.Mean calibration trivially avoids the separation problemby not calibrating the transmitters at all. Whil
29、e this sys-tem minimizes expected error for the receiver, however, itis unlikely that it minimizes error for the system as a wholesince the transmitters have not been calibrated. In the nextsection, we describe a method that directly minimizes theerror of the entire system.5. CALIBRATION IN CALAMARI
30、Both methods that we have seen so far are micro-calibrationprocesses. They directly observe each device and build amapping from r to rto directly optimize that devices re-sponse. In this section, we will describe a method of macro-calibration that calibrates each device by optimizing theoverall syst
31、em response instead of the individual device re-sponses.The method has three steps:1. Parameterize each individual device and model thesystem response as a whole using these parameters2. Collect data from the system as a whole3. Choose the parameters for the individual devices suchthat the behavior
32、of the entire system is optimizedHow does this technique choose parameters for individ-ual devices while only observing the system response? Byobserving trends in the transmitter/receiver pairs, we canattribute errors in the system to the individual nodes thatare likely to cause them. In other words
33、, if all distance esti-mates made with a particular transmitter are slightly high,we can blame that transmitter. By looking at the entiresystem response at once, we can allocate blame optimallyamongst the nodes.5.1 The ParameterizationWe choose parameters for the devices based on our phys-ical under
34、standing of their interactions. We focus here on61 TOF ranging, although the parameterization is nearly iden-tical for RSSI.In Calamari, an acoustic pulse of roughly 15ms is trans-mitted along with a 25ms radio packet. When the micro-phone receives the acoustic pulse, the phase lock loop of thetone
35、detector locks onto the signal. When the PLL responseand the microphone response are high enough in combina-tion, an interrupt is fired which is time-stamped by the pro-cessor. This time stamp is compared with the time stampof the radio packet. The dierence in time is multiplied bythe nominal speed
36、of sound to obtain a distance estimate.Several variations in the hardware strongly eect the TOFreadings.1. The time for the sounder and microphone to start os-cillating may vary due to variations in the diaphrams2. The volume of the sounder and the sensitivity of themicrophone aect the speed with wh
37、ich the PLL de-tects the signal3. A dierence in sounder frequency and tone detectorscenter frequency reduces the speed with which the PLLdetects the signal4. The relative orientations of the sounder and micro-phone may aect the volume with which the acoustictone is receivedWe therefore arrive at the
38、 following complete model of thesystem response for a transmitter/receiver pair:d= BT+ BR+ GT d + GR d+|FT FR|d + fO(OT,OR) d (6)Where d is the distance estimate and disthetruedis-tance between the pair. BTand BRrepresent the startuptimes for diaphram oscillation, which aect the observedTOF by a con
39、stant. GTand GRrepresent transmitter vol-ume and receiver sensitivity, respectively, which have an af-fect proportional to the distance. |FT FR| represents theeects due to frequency dierences, and fO(OT,OR)repre-sents the eect on volume due to the relative orientations ofthe two devices.5.2 Choosing
40、 the ParametersAssuming we have collected data from our system, we maychoose values for the device parameters such that our sensormodel above matches the data we collected. To simplifythe process, we will remove the non-linear parameters forfrequency and orientation and allow both of these factors t
41、obe built into the error term. Our simplified but less accuratesensor model becomes:r= BT+ BR+ GT r + GR r (7)Each pair of collected values r and rtogether with thismodel form an equation with four variables. We collect onesample from each pair in our network, giving us 4n variablesin n2 n equations
42、, where n isthenumberofnodes.Thissystem of equations will generally overconstrain the valuesof all device parameters.We use least-squares 2 14 to do exactly that by con-verting our system of equations to matrix notation havingthe following form:Ax = b (8)Let x be an array of our device parameters an
43、d let eachrow of A be the coecients from our model of one datasample b. For example, if our first two distance estimateswere d1,2and d1,3coming from transmitter/receiver pairs(1, 2) and (1, 3), the first two rows of the A and b matriceswouldbeasfollows:A =10 010 d1,20 0 d1,2010 001 d1,30 00d1,3.andb
44、 =d1,2d1,3.wherex =BT1.BTnBR1.BRnGT1.GTnGR1.GRnCollecting k distance estimates from our system, we createthese matrices and solve for x,whereA is a k 4n matrix,b is a k 1matrixandx is a 4n 1 matrix.As you can see, this method only uses the estimated dis-tances given by the transmitter/reciever pairs
45、. The actualresponse of any particular transmitter or receiver was neverobserved yet every transmitter and receiver has its own pa-rameters. One big advantage to this method is that any newnodes added to the network will not have to be pairwise cal-ibrated with each existing node but only with a sam
46、ple oftransmitters and receivers.In the next two sections we empirically evaluate this methodand compare it with the methods previously described.6. EXPERIMENTAL SETUPIn this section we describe the exact hardware and method-ology used to collect data and evaluate these calibrationmethods.Calamari i
47、s built on the MICA sensor platform 11. TheMICA mote is essentially an Atmel 103 microcontroller inconjunction with a RFM TR1000 radio transceiver and isabout the size of two AA batteries, as seen in Figure 3.Connected to each mote is a MICA sensor board 16 which,among other things, contains a Siriu
48、s PS14T40A 4.3KHzsounder and a Panasonic WM-62A microphone, whose band-passed output is wired to a National Semiconductor LMC567CMtone detector whose center frequency is set to about 4.5KHz.62 Figure 2: Experimental TestbedFigure 3: Mica sensorboard mounted on a moteIn our experiments we use 32 node
49、s in a 30cm x 30cmgrid. The motes are situated atop a large table in an 8x 4 formation spanning a total area of 210cm x 90cm, asseen in Figure 2. All motes are in the same orientation.Both the sounder and microphone point directly upwards.The sounder is located in the center of the board while themicrophone is in one corner, as seen in Figure 3.Although this experimental setup does not test all combi-nation
