1、Inexpensive and Automatic Calibration forAcceleration SensorsAlbert Krohn, Michael Beigl, Christian Decker, Uwe Kochendorfer, Philip Robinson and Tobias ZimmerTelecooperation Office (TecO)Universitat KarlsruheAbstract In this paper, we present two methods for calibra-tion of acceleration sensors tha
2、t are inexpensive, in-situ, requireminimum user interaction and are targeted to a broad set ofacceleration sensor applications and devices. We overcome thenecessity of orthogonal axes alignment by extending existingcalibration methods with a non-orthogonal axes model. Ournon-orthogonal method can fu
3、rthermore be used to enableautomatic calibration for 1- or 2-axes accelerometers or realizea simultaneous mass-calibration of sensors with minimum effort.The influence of noise to the presented calibration methods isanalysed.I. INTRODUCTIONCalibration is an important issue in sensor based systemsas
4、it is the only way to ensure a predictable quality ofdelivered information. Traditionally, calibration is done in aprocess during production time. The process is very costly andsometimes requires manual steps which makes it difficult tolower costs at this point. Recalibration is necessary after some
5、time for most sensor systems and can often only be done atparticular sites and hence incurs additional high costs. Ubiq-uitous computing environments differ in many aspects fromtraditional sensing settings. Ubiquitous Computing devicesare used as peripheral devices disappearing from the usersawarene
6、ss. Therefore it can be said that the majority of Ubiq-uitous Computing devices will be factually and metaphoricallyinvisible 1. As a consequence, users tend to ignore intensedevice administration, even though operational conditions maybe extremely rigourous, such that errors due to neglected recal-
7、ibration accumulate. Furthermore, the circumstances of devicedeployment in Ubiquitous Computing environments typicallysees them embedded into other objects, hindering access andopportunity to perform mechanical recalibration. The amountof computational devices deployed is also envisioned to beon a h
8、igher scale in comparison to traditional sensor systems.We refer to hundreds of devices spread in the environment,comparable to the random distribution of low costs, everydayobjects and consumer electronics to date. This introduces newchallenges for management and maintenance especially for thecalib
9、ration and recalibration processes.In research, this development has already started with theinstrumentation of everyday objects such as cups 2. Inorder to allow embedding, the devices must be very small,priced in the cent range and absolutely maintenance free.Costly calibration including high price
10、d hardware is not anoption for these applications. Calibration has to be done in-situ without user intervention and additionally has to be doneunder the assumption that the sensor technology is cheap. Awidely used sensor technology in Ubicomp is the MEMS-type acceleration sensor. The sensors are sma
11、ll and quiteaccurate. Various platforms and applications that incorporatesthem have been developed. Examples are Lancasters DIYSmart-Its 3, TecOs particle computers 4,the WearPen andTiltPad 5. Additionally, companies like Crossbow 6 andSilicon Designs 7 offer products for easy integration ofaccelero
12、meters in other products even for 3-axial accelerationmeasurements.Fig. 1. 3-Axes accelerometer built out of 1-axis sensors c Silicon Designs7However, calibration is necessary, especially if accelerationsensors are used collaboratively. For example, if a 2-axes ac-celeration sensor device and a seco
13、nd 1-or 2-axes accelerationsensor device are used together in one system - e.g. attachedto the same object - both devices can spontaneously forma 3-axes acceleration sensor device. Many of the mentionedplatforms build their 3-axes acceleration sensors out of two2-axes acceleration sensors that are o
14、rthogonally mounted.One example is the mentioned design from Silicon Designs.Figure 1 7 shows how three 1-axis sensors are mechanicallycombined to a three-dimensional acceleration sensor.In this paper we present a method that allows us tocalibrate 3-axes accelerometers simply and simultaneously.It a
15、lso makes it possible to combine several 1- or 2-axesaccelerometers and to form a cheap 3-axes acceleration sensoron the fly and then calibrate the underlying 1- or 2-axessensors. Such calibration can be done in parallel for manycombined 3-axes sensors:Hundreds of these sensor devices can be calibra
16、ted simulta-neously by just putting them into one box and calibrating themall with only some measurements in different orientations ofthe box. During the measurements, the 1- and 2-dimensionalsensors would build 3-axes accelerometers by virtually com-bining them to 3D sensor systems. The main proble
17、m to dealwith in these setting is that the axes of the combined sensorsare not orthogonal but randomly oriented in the box. Thisproblem is not restricted to simple and cheap sensors it is alsoa problem for most off-the-shelf 3 dimensional accelerationsensors. The axes of the sensors are often not pe
18、rfectly alignedto 90 due to mechanical impreciseness. The model that ispresented in section IV presents a solution for such 3D sensorcalibration of sensors that have non-orthogonal axes either ifthey are already assembled or are assembled on-the-fly to forma 3-axes acceleration sensor.One approach f
19、or in-situ calibration of 3-axes accelerationsensors involving the parameters offset and scale was intro-duced by Lukowicz et al. in 8. They proposed a method forthe calibration that needs only some random measurements indifferent orientations taking advantage of the earths gravityfield.Our paper fo
20、cuses on in-situ calibration methods espe-cially regarding noise in the measurements and tilted (non-orthogonal) accelerometer systems. We will extend the currentapproaches in order to take care not only of offset and scalewithin the calibration, but also of the orthogonality of the axes.The charact
21、eristic of in-situ calibration will be kept.The paper proceeds with a short overview of calibrationmethods involving only offset and scale. Section III investi-gates the influence of measurement noise and tilted axes on themethods. Section IV deals with non-orthogonal axes and givesan extension of t
22、he traditional approaches. Implementationconsideration are covered in section V before we concludein section VI.II. STATE OF THE ART CALIBRATIONIn many applications there is a need for a calibrated accel-eration measurement, but many sensors - especially MEMStypes - are not calibrated after producti
23、on. Instead the sensorhave a sensitivity s and an offset o on each of their axes whichlead to measurements that do not represent the actual physicalvalue. Before presenting the two most important calibrationtechniques, we define some expression used throughout thepaper: the physical acceleration is
24、named as x = (x;y;z)T . Itis measured in multiples of the earths gravity g the measurement offsets of the axes of sensors are namedox;oy;oz and the scalings sx;sy;sz, whereas a perfectsystem would have o =0 and s =1 the values measured by an uncalibrated, orthogonal 3Dacceleration sensor are named u
25、 = (u;v;w)T ; uncali-brated means here that offset and scaling errors are stillpresent in the measurements the values measured by a calibrated, non-orthogonal 3Dacceleration sensor are named r = (r;s;t)T ; calibratedmeans here that offset and scaling errors have been dis-covered and the measurements
26、 are corrected accordingly the interrelationship between measurements and realphysical values in an orthogonal system is for x-axis:x = (u ox)=sx;. For the y- and z-axis: the accordingequations.A. Rotational CalibrationThe method is described in 9 and determines the offsetand the scale factor for ea
27、ch axis separately. Hereby, an axis(e.g. the x-axis) of the acceleration sensor is oriented to theearths gravity centre and kept stationary. It is exposed to 1gand rotated and exposed to -1g. The measured values (in g)in both positions are umax;x and umin;x. Solving the equationsystem1 = umax;x oxsx
28、1 = umin;x oxsxwill result in the offset ox and scale factor sx for this axis:ox = umax;x +umin;x2 (1)sx = umax;x umin;x2 (2)In order to find umax and umin the rotation has to be carriedout very slowly to minimize the effect of dynamic accelerationcomponents. The accuracy of the method relies signif
29、icantlyon the accuracy of the alignment.B. Automatic CalibrationAnother method for calibration of 3-axes accelerometers hasbeen presented in 8. It does not require a certain series ofpre-defined positions like the method explained above and istherefore very practical in mobile settings. It can perfo
30、rm acalibration in-situ after a complete device has been assembled.It is also suitable for fast mass-calibration as the sensorvirtually calibrates itself. The idea is to use the earths gravityforce as a known static acceleration on a 3-axes accelerometerwhen a sensor has no dynamic component applied
31、 on it. Is thisstate of being stationary detected the following equation (see8, page 2) is valid:jxj =p(x2 +y2 +z2) = 1 (3)With the according offsets and scale errors of the accelerom-eters, equation (3) extends to:(uox)=sx)2 +(v oy)=sy)2 +(woz)=sz)2) = 1 (4)With equation (4) the six unknown calibra
32、tion variables(ox;oy;oz;sx;sy;sz) can be solved when creating a six linedequation system using six earths gravity vectors measuredin different orientations of the sensor. The six necessarymeasurements should be significantly different from eachother to guarantee a stable convergence of the non-linea
33、rsolver. Additionally, it must be assured, that all three axes areperfectly aligned to be orthogonal, otherwise the preconditionfor the algorithm is violated and the results will be errorneous.C. Concerns Using the Presented Calibration MethodsWhen calibrating acceleration sensors in MEMS technology
34、we generally experience the problem of noise introduced in themeasurement. Noise can have different reasons like thermalnoise, quantization noise or noise introduced by A/D conver-sion. Noise during the calibration corrupts the measurementsand therefore results in imprecise calibration. It is necess
35、aryto have a quantitative insight of the influence of noise to acalibration process.Further, the calibration of the axes angles is not addressedin either calibration method. A complete calibration of a 3Daccelerometer includes as well the calibration of the axesangles to 90 to be able to calculate t
36、he physical acceleration xfrom the measured value u. Some settings (mentioned in sec-tion I) show that its necessary to provide calibration methodsfor tilted axes. Furthermore, the axes angles calibration is ofspecial interest for the automatic calibration method as theorthogonality is the precondit
37、ion for the validity of equation(3). With titled axes, the automatic calibration method will notwork!III. INFLUENCE OF NOISEIt is impossible to avoid errors on the calibration dueto noise during the calibration process. To figure out whatnegative influence on the calibration results must be expected
38、from a known noise level, we simulated different noise levelswith an even distribution and a maximum of 5, 10, 20 and 50mg and applied them on virtual measurements before usingthem in the two calibration procedures. Figures 2 and 30 50 100 150 2000102030405060708090100offset error mgaccumulated perc
39、entage of occurrencenoise 5mgnoise 10mgnoise 20mgnoise 50mgnoise 5mg, tilted axesnoise 20mg tilted axesFig. 2. Offset errors for simulated automatic calibrationshow the accumulative error distribution of the calibrationparameter offset using the two presented calibration methods.The offset errors ar
40、e shown in absolute error, the scaling errorsin Figures 4 and 5 relatively. Reading out from Figure 2 withparameter noise = 20mg at abscissa offseterror = 50mggives the value 75. That means that statistically for 75% ofthe calibrations that have a 20mg noise level, the offset error0 10 20 30 40 5001
41、02030405060708090100offset error mgaccumulated percentage of occurrencenoise 5mgnoise 10mgnoise 20mgnoise 50mgFig. 3. Offsets errrors for simulated rotational calibrationin the results of the calibration will be less or equal 50mg.The 75% can be regarded as the statistical trust level of theoffset e
42、rror being less or equal 50mg in the calibration whenapplying 20mg of noise on the measurement.A quick example should show how the Figures 2 and 3 forthe offset errors can be compared to Figures 4 and 5 for thescaling errors: An error of 10% in scaling would cause an errorin a measurement of approx.
43、 100mg when stationary pointingtowards ground. Therefore, as a rule of thumb, 1% error inscaling can be compared to 10mg in offset error. The graphicsare useful when a certain target accuracy is desired. Then, onecan read out what the (mean) necessary input accuracy shouldbe. For example, a desired
44、target accuracy of 50mg in offsetand 5% in scale with a trust level of 95% would require anoise level around 5mg using the automatic calibration.0 10 20 30 40 50405060708090100scale error %accumulated percentage of occurencenoise 5mgnoise 10mgnoise 20mgnoise 50mgFig. 4. Scaling errors for simulated
45、automatic calibrationWhen comparing e.g. the 10mg noise curves of Figure2 and 3, one can see that the 95% confidence level forrotational method gives an offset error of around 100mg wherethe rotational method reaches 8mg! On the first sight, therotational method would be the clear winner. But the re
46、sultneeds to be interpreted in a broader context. The solver ofthe non-linear equation array causes additional errors on theautomatic calibration. And the noise model on the rotational0 1 2 3 4 5 6 7 80102030405060708090100scale error %accumulated percentage of occurrencenoise 5mgnoise 10mgnoise 20m
47、gnoise 50mgFig. 5. Scaling errors for simulated rotational calibrationalgorithm does not include the errors that occur when thesensor was not measured in its minimum and maximum butslightly off to these points. In 8 the authors gain a significantimprovement in accuracy by averaging a series of resul
48、ts ofone sensor.IV. NON-ORTHOGONAL AXESAs already mentioned, the calibration of axes is not sup-ported by either presented calibration method. For the rota-tional calibration, the axes are calibrated independently andthe offset and scaling calibration is not negatively affected byangular displacemen
49、t. For this reason, we propose an extensionof the rotational algorithm in section IV-B to calibrate possibleaxes displacements as a second step after the standard offsetand scaling calibration.For the automatic calibration method, it is a necessary pre-condition that the axes are orthogonal. We simulated somecases, where the angle was off by 5. These curves can befound as dotted lines in Figure 2. The error is significantlyworse than without angular displacement, which motivatesthe extension of the algorithm to
