Intelligent Calibration Method of low cost MEMS Inertial ...Intelligent Calibration Method of low...

International Journal ofIntelligent Engineering & Systems

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Intelligent Calibration Method of low cost MEMS Inertial Measurement Unitfor an FPGA-based Navigation System

Lei Wang1∗, Fei Wang2

1 Center of Micro-system Technology, Shenyang Ligong University, Shenyang 110159, China2 Mechanical Engineering & Automation College, Northeastern University, Shenyang 110004, China

∗ Corresponding author’s Email: [email protected]

Abstract: Based on a small, lightweight, low-cost high performance inertial Measurement Units(IMU), an effec-tive calibration method is implemented to evaluate the performance of Micro-Electro-Mechanical Systems(MEMS)sensors suffering from various errors to get acceptable navigation results. A prototype development board based onFPGA, dual core processor’s configuration for INS/GPS integrated navigation system is designed for experimentaltesting. The significant error sources of IMU such as bias, scale factor, and misalignment are estimated in virtue ofstatic tests, rate tests, and thermal tests. Moreover, an effective intelligent calibration method combining with Kalmanfilter is proposed to estimate parameters and reduce the effect of IMU dynamic errors that can degrade the systemperformance. The efficiency of proposed approach is demonstrated by various experimental scenarios.

Keywords: MEMS IMU; Intelligent Calibration Method; Kalman Filter; FPGA; Navigation

1. Introduction

In recent years, a promising technology, micro-elec-tro-mechanical systems (MEMS)-based inertial sen-sors, has been developed which can provide a low-costnavigation solution. MEMS systems are commonlyfabricated using silicon, which possesses significantelectrical and mechanical advantages over other ma-terials [1, 2]. With the rapid growth in demand, suchas in applications of general aviation, unmanned au-tomotive vehicle, personnel localization, mobile map-ping systems, athletic training monitoring and com-puter games, etc [2, 3].

In addition to advances in MEMS fabrication tech-niques, one of the primary drivers behind the phe-nomenon of high performance MEMS inertial sensorsis the skill and ability of the MEMS integrator to pack-age these devices into an Inertial Measurement Unit(IMU) [4]. Today’s MEMS sensors are still muchless precise than expensive accurate inertial sensors,

such as tactic or navigation grade IMU whose mea-surements are able to be directly used by inertial sys-tem self-alignment and strapdown inertial navigationalgorithm. In principle, an IMU consists of Tri-axisMEMS gyroscopes and accelerometers which mea-sure angular velocities and accelerations in three di-mensions. However, if applied by modern MEMSsensors, these standard inertial calculation proceduresare not practical, or in other words, the solutions di-verge quickly. The large MEMS noises cause the stand-alone use of MEMS sensors in strapdown inertial nav-igation system (INS) to deliver kilometer level posi-tioning errors for the applications of several secondsduration [3]. Therefore, there is a critical issue whichfocuses on finding the solution to improve the perfor-mance of IMU based MEMS sensors.

IMU errors include misalignment errors, biases, andscale factor errors, etc [5]. Low cost sensors, such asMEMS sensors, are quite large, and their repeatabilityis typically poor because of environment factors, espe-

International Journal of Intelligent Engineering and Systems, Vol.4, No.2, 2011 32

cially temperature, which makes frequent calibrationa necessity. Different calibration techniques devel-oped by many authors differ mainly in the proposedsensor output model, the used instrumentation, andthe thermal compensation.

Bekkeng [5] carried out a calibration analysis of gy-roscopes. Gyro parameters had been estimated throughkalman filter. Aggarwal [6] proposed a standard test-ing/calibration procedure for MEMS inertial sensorsand a linear thermal model is developed to compen-sate for the effects of errors due to the temperaturevariation. Fang [8] analyzed the dynamic thermal er-ror induced by accelerations of MEMS gyroscope. Ot-her researchers [10, 11] proposed the linear interpola-tion, 3rd order polynomial, generic hysteresis model,and characteristic thermal error compensation meth-ods.

Most previous works only focus on thermal com-pensation of IMU in reference to static environment.Sometimes the compensation is indispensable in ap-plications where there are no static periods. There-fore, the variation of real-time biases, scale factorsin low cost MEMS sensors should be taken into ac-count in the integration of dynamic rate and temper-ature variation simultaneously. Moreover, there is aneed for compensating dynamic noise in addition tothe static noise considering the effect of temperaturevariation in low cost MEMS IMU.

This paper focuses on the analysis and compensa-tion of low cost and accuracy IMU error induced bydynamic environmental variation of temperature andrate. Due to the variation of IMU errors irregularlypresent in dynamic environment, such as dynamic biaswith time, scale factor, the conventional compensationapproaches are not effective to improve the compen-sation accuracy of low cost MEMS IMU. An intelli-gent calibration approach based on combination withfuzzy control, artificial neural net work with a Kalmanfilter is proposed. Taking advantage of the integratedcompensation method, the thermal error in static testis compensated by measuring the temperature. On theother hand, the random error, time-varying bias in ratetest is estimated by Kalman filter on line. The accu-racy of inertial sensors can significantly be improvedby performing precision calibration of each sensor. Inthis way, the performance of calibrated MEMS IMU isimproved and can be used for stabilization, guidanceand inertial navigation.

The paper is organized as following: a standard cal-ibration and testing method is developed to determineerrors. Furthermore, according to the MEMS IMU

and INS/GPS technology, a prototype based on FPGAdevelopment board, dual core processor’s prototypeof integrated system is developed and researched forconventional navigation. The platform integrating ME-MS IMU/GPS/INS provides superior performance thanany of the methods operating alone. The System-On-chip Circuitry (SOPC) implementation was developedwith VHDL language into an FPGA, so as to be easyto evaluate the customized IMU error compensationand implement the navigation algorithm.

The proposed intelligent calibration method uses mea-surement results to estimate parameters and eliminatethe effect of IMU errors in various temperatures andrates. The temperature compensation is performed toremove the thermal sensitivity of the MEMS IMU.Experimental results are afforded and evaluated.

2. Calibration method and validation

2.1 Inertial sensor error modelTraditionally, the inertial sensor errors can be di-

vided into two parts, deterministic (systematic) errorsand random errors [6, 7]. The random errors includebias-drifts or scale factor drifts. These errors changewith time and dynamic rates. These random errorshave to be modeled stochastically. The deterministicerror sources include bias, non-orthogonality or mis-alignment errors and scale factor errors which can beremoved by specific calibration procedures in a labo-ratory environment. It is demonstrated that the deter-ministic error is temperature dependent [8].

An inertial sensor error model is developed in thissection. It is a unified model because it applies to theboth accelerometers and rate gyros. A general modelfor the error of a single inertial sensor is as follows:

sm = K · st +b(t) (1)

where sm is the measured quantity at the sensor out-put and st is the true value of the quantity that sensormeasures. The term K represents scale factor and b(t)is the output bias.

The total output bias b(t) is the residual output whenno input is applied. Thus, it can be measured when thesensor is static. The total bias represents an additiveerror and consists of several components in (2),

b(t) = b0 +br(t)+bw(t) (2)

where b0 is a constant null-shift, br(t) is a slow timevarying component called the bias drift, and bw(t) isthe random component known as noise. It is straightforward to determine numerical values for the error


terms b0 and bw(t) . The noise bw(t) can be modeledas a band-limited white noise. A numerical value forbw(t) can be determined by computing the standarddeviation of the sensor’s output over a short periodof time when no input is applied. The time varyingcomponent of the output error, br(t) , is character-ized by a stochastic time series. Characterizing thisslowly varying process by a single number would beoverly conservative in the short term and would notadequately model the longer term error variation [7].

This suggests modeling the time varying bias com-ponent will be modeled as a first order Gauss-Markovprocess. This process can be described by a first orderdifferential equation.

br =−1τ

br +bg (3)

Where τ is the correlation time constant and bg is aGaussian white process with variance E[br

2(t)] = σr2

and time constant τ have the following properties, Ex-ponential autocorrelation

Rx(τ) = σr2e−

tτ (4)

In lab, we developed a MEMS IMU which consistsof three ADXR150 gyros and three ADXL210 accelerom-eters. The constructed MEMS IMU, the unit dimen-sions are (3.3×3.3×4)cm in Figure 1.

Figure 1 Photograph of MEMS IMU and FPGA prototype

The Allan variance method [9] can be used to de-termine br(t) in (2). By constructing the Allan vari-ance plot, the slow varying process corrupting a sen-sor output can be characterized. For stochastic inertialerror, the Allan variance log-log plots for gyro and ac-celerometer are shown in Figure 2 and Figure 3.

Figure 2 shows the Allan variance plot constructedfor the device ADXR150 gyro output. The analyzed

data set consists of gyro measurement at a samplingfrequency of 400 Hz. The Allan variance has an ap-proximate slope of -1/2 for the first 35 seconds. Thisindicates that the angular random walk(ARW) is thedominant noise term. After 35 seconds, the Allan vari-ance has a slope of +1/2 which implies that the veloc-ity random walk (VRW) is the dominant noise term.

Figure 2 AV result for the device ADXR150 gyro

Figure 3 AV result for the device ADXL210 accelerometer

The VRW and ARW values are calculated. Theseparameters are required for design the process noisematrix Q to be used in Kalman Filter algorithm in theintegrated navigation systems. The same analysis pro-cess in Allan variance plot is constructed for the de-vice ADXL210 accelerometer output.

2.2 MEMS gyro and accelerometer model in labGyroscopes fabricated by micro electromechanical

system (MEMS) technology offer revolutionary im-provements in cost, size, and ruggedness relative tofiber-optic and spinning mass technologies. The gyro


model is used for calibration process in MEMS IMUin lab. It is a little different with the unified inertialsensor error model in Equation (1).

Three orthogonally mounted ADXR150 rate-gyrosin order to measure angular rates around the sensi-tivity axes. Practically, the sensors are not preciselymounted orthogonally due to imperfections during theIMU construction process. Therefore, there will bea small angular position differences between inertialsensors sensitivity axes and the IMU frame or plat-form in which inertial sensors are mounted, which iscalled misalignment errors or non-orthogonalities asshown in Figure 4.

Figure 4 IMU platform coordinate axes, inertial sensor sen-sitivity axes

The stabilized platform (xtytzt) coordinates can bedefined to be the orthogonalized sensor input axis co-ordinates. The coordinate frame (xmymzm) representsthe non-orthogonalized frame of actual sensitive axesof MEMS IMU. The offset θ implies the transforma-tion from the non-orthogonal sensor input axis clusterto the platform orthogonal coordinates with the sameorigin. θi j(i, j = x,y,z) is the rotation of i-th sensorsensitivity axis around j-th platform sensitivity axis.In matrix form the output of a triad of gyro sensorscan be represented as follows:

ωmxωmyωmz

=

bxbybz

+

dxx dxy dxzdyx dyy dyzdzx dzy dzz

·

fmxfmyfmz

+

Kωx 0 00 Kωy dyz0 0 Kωz

·

CxzCxy SxzCxy SxySyz CyxCyz SyxCyz

SzyCzx Szx CzyCzx

·

ωtxωtyωtz

(5)

where ωm j( j = x,y,z) is gyro sensor measurement;ωt j( j = x,y,z) is the true value of the quantity thatgyro sensor measures; b j( j = x,y,z) is the bias outputof gyro; di j(i, j = x,y,z) is the gyro drift parameters

depending on acceleration; and Kω j( j = x,y,z)is thescale factor. The terms cosθ = C,sinθ = S are sub-stituted in formula (5), which represents the misalign-ment errors.

The approximate derivation process is used in MEMSaccelerometer model. In matrix form the output of atriad of accelerometer sensors can be represented as

ωmxωmyωmz

=

Kax 0 00 Kay 00 0 Kaz

·

CxzCxy SxzCxy SxySyz CyxCyz SyxCyz

SzyCzx Szx CzyCzx

·

ftxftyftz

+

baxbaybaz

(6)

where fm j( j = x,y,z) is accelerometer sensor mea-surement; ft j( j = x,y,z) is the true value of the quan-tity that accelerometer sensor measures; ba j( j = x,y,z)is the bias output of accelerometer; Ka j( j = x,y,z)is the scale factor. In accelerometer model, the ac-celerometer drift parameters depending on rotation arenot considered in most cases.

2.3 IMU calibration testing methodThe calibration of IMU is required to reduce the er-

rors in the INS derived position, velocity and attitudeof moving platforms. To get rid of the inertial error asmuch as possible, the modified methods used for cal-ibrating IMU were primarily designed for static andrate tests. The improved calibration method combin-ing with a static test, the dynamic rate test and thermaltesting is implemented.

The standard six-position test method [6] requiresthe inertial system to be mounted on a fixed surfacewith each sensitivity axis pointing alternately up anddown. In our tests, besides the above experiments,additional four positions are considered. Especially,the IMU is mounted on inclined plane in precise fixedangle in static test in Figure 5. It is called ten-positionstatic method.

Figure 5 The constructed MEMS IMU in inclined plane

In order to find the scale factor and the bias at leasttwo pairs of points of the calibration procedure are


needed. Accelerometers are sensitive to gravity ac-celeration, so when each one of its sensitivity axes isplaced parallel to gravity in both positive and nega-tive senses, the measured accelerations will be 1g and-1g respectively. Therefore, two pairs of points areobtained for each axis just by placing the IMU on afix surface leaning on the two perpendicular sides tothe gravity axis, i.e, a 180◦ rotation is done along thethree accelerometer axes.

In the lab, the adopted ten-position static test com-prises following steps. The sensitivity axes of IMU(x,y,z) is placed in the upward-vertical direction of thelocal level frame alternately. Later these sensitivityaxes of IMU are rotated 180◦ around upward-verticalaxis of the local level frame alternately. Through theabove procedures, six static positions are located forIMU. Furthermore, among sensitivity axes of IMU,any axis is placed in the downward-vertical directionof the local level frame and it is rotated 180◦ arounddownward-vertical axis of the local level frame to cho-ose two static positions. The two different sensitivityaxes of IMU are placed with 30◦ inclined angle forthe last two positions. There are ten position of IMUaltogether.

By placing the IMU in reference to standard up-down four position method, the accelerometers mea-surement fm will be

+g00

,

0+g0

,

00

+g

,

00−g

(7)

In order to estimate the bias, scale factor, non-ortho-gonalities parameters in gyro error model, not onlystatic position test, but also rate test for gyro is indis-pensable. Rate tests are typically done using a pre-cision single-axis rate table. The rate test for MEMSIMU is shown in Figure. 6.

In static test, the true value of quantity ωt j( j = x,y,z)for gyro is determined as follows:

ωtx

ωty

ωtz

=

ωiε cosϕ sinψn

ωiε cosϕ cosψn

ωiε sinϕ

(8)

where the term ϕ is the local latitude; the term ψnis the angle between y-axis of IMU and north direc-tion in inertial navigation coordinate system. The bi-ases of the gyroscope can be estimated using the sameprinciple as the elimination method of symmetricallyaligned position. For example, if the y axes of IMUare rotated 180◦ around upward axis of the local levelframe, the value of ψn would be changed. The corre-sponding relationship would be obtained as follows:

sin(ψn +180)=−sin(ψn),cos(ψn +180)=−cos(ψn) (9)

Figure 6 Rate rest for MEMS IMU

In rate test, the true value of quantity ωt j( j = x,y,z)for gyro is as follows:

ωtxωtyωtz

=

ωiε cosϕ sin(ψn +ωrx · t)ωiε cosϕ cos(ψn +ωry · t)

ωiε sinϕ

+

ωrxωryωrz

(10)

where ωr j( j = x,y,z) is the defined rotating rate ofrate platform. The t is running time of rate table. TheIMU is mounted on a rate table of single. Each sensi-tivity axis of IMU is placed in direction rotating axisof rate table alternately. The term will be

+ψ j00

,

0+ψ j

0

,

00

+ψ j

,

−ψ j00

,

0−ψ j

0

,

00

−ψ j

(11)

where, ψ j is the numerical value of at the jth rotatingrate. The positive and negative rate tests are accom-plished through an amount of different defined anglerates in both the clockwise and counter clockwise di-rections. According to the test data, the scale factorand non-orthogonality parameters are decoupled andestimated through combination the static and rate testcalibration method. The estimation techniques of leastsquares and cyclic iteration are developed to solve thecoupling of scale factor and non-orthogonality in Equa-tion (5). Simultaneity, combining the rate test withstatic test, the adverse infection of the earth rotationrate could be eliminated in solving the equation, con-sidering the reason that low grade MEMS sensors suf-fering from bias instability, which can completely maskthe earth’s reference signal. Table 1 lists the main cal-ibration results in Gyro ADXR150.

Table 1 Gyroscope deterministic errors for ADXR150Axis Gyro Deterministic error in indoor Temperature

Biases (◦/s) Scale factorGyro x 19.81 0.988Gyro y 15.469 1.019Gyro z 13.7 1.027


Figure 7 The circuit of FPGA Navigation computer (9×6×2cm)

2.4 FPGA based navigation prototypeThe testing platform of IMU is a prototype GPS/INS

integrated navigation system, built on an Altera’s FPGAdevelopment board. The prototype aims at a low-cost,small-size, and portable navigation system for guid-ance application. The integrated navigation systembased on FPGA is shown in Figure 7.

The hardware’s core of FPGA board is dual NIOSIIsystem, which consists of two NIOSII processors. A-mong the two processors, the main processor is re-sponsible for customized navigation program designand calculation task, and the coprocessor guaranteesthe communication ability of I/O processor and MEMSIMU/GPS, calibration module, etc. The System-on-chip circuitry (SOPC) implementation was developedwith VHDL language into an FPGA, so as to be easyto compensate the MEMS IMU error and implementthe navigation algorithm. Calibration testing experi-ments are implemented to improve the performanceof prototype integrated system based on FPGA. Notonly data processing in system, but also real-time ofcommunication are realized.

Some experiments are implemented to test the per-formance of prototype GPS/INS integrated system ba-sed on FPGA. The processor read the data of signal ofinertial sensor at 10ms sampling time. It costs 10∼15msto accomplish the computation of INS navigation. Thecorresponding parameters could be transferred to PCterminal or other utilities in 10ms.

3. Intelligent calibration method combiningKalman filter

3.1 Intelligent thermal calibration methodBias drift is probably the most crucial of the inertial

errors. In inertial sensors even in the absence of anyinput (acceleration or angular velocity) the output isnon-zero. This offset, which is usually referred to asbias, is added to the actual measured signal. This ini-

Figure 8 Thermal testing for MEMS IMU

tial deterministic IMU error calibration does not guar-antee proper measuring during all the time the MEMSIMU is activated since start-up. Unluckily MEMSIMU has a strong temperature dynamic bias, whichmeans the bias will vary when temperature changes.There is a need for the development of accurate, reli-able and efficient thermal models to decrease the ef-fect of these temperature variations on IMU errors [6].To investigate thermal effect of sensors and to evaluatepiecewise local temperature drift compensation mod-els, thermal testing for IMU is performed in thermalchamber in static environment. It is shown in Figure8.

In most cases, the main source of temperature varia-tion is not ambient temperature but circuit selfheating.Such biases are separated into two parts. One is stabi-lized bias with a fixed temperature, the other is a smalltime varying bias.

In order to evaluate the stabilized biases and thescale factor values for the accelerometer and gyro-scopes, a local linear interpolation was performed [11].Considering the variation largely of bias in MEMSIMU in full temperature, this method will be less reli-able. In allusion to the defect of descension of thermalcompensation method mentioned above, an intelligentthermal compensation combining the artificial neuralnetwork and fuzzy control(F-NN) is proposed in thispaper. Thermal compensation experiments for stabi-lized biases are implemented in temperature chamberin static environment.

By virtue of the measurement data of IMU in fulltemperature on line, the characters and variation rulesof stabilized biases in full temperatures is establishedthrough an offline training of feed-forward neural net-work. After that the adaptive neural network is imple-mented for online compensation of affection of theseerrors. During the process of training and learning, the


learning rate of Network is regulated by fuzzy logicrules in order to accelerate the converging process oflearning and improve the ability of adaptation. Theback propagation update rule for the weights with amomentum term is as follows:

∆wi j(k +1) = η ·G(k) ·ζ +α∆wi j(k) (12)

where η is the learning rate of neural network, α is themomentum coefficient and G(k) is the negative gradi-ent, ∆wi j is weights of adjusting, ζ is output of neu-ron.

The input of NN is the value of temperature of thethermal chamber, the output of NN is the correspond-ing variation of stabilized biases in full temperaturetesting. The non-linear relation between temperatureand stabilized bias is determined based on NN. More-over, the learning rate of NN is adjusted in trainingprocess. Define the Standard deviation of NN is E,and the change of Standard deviation is EC. The in-put to the fuzzy controller is the Standard deviation Eand the change of Standard deviation EC, which rep-resents the magnitude of error of desired and actualoutput of NN, and its output is the learning rate η .

The following tuning rule is constructed for the learn-ing rate η . When Standard deviation E and its changeEC vary largely, larger changes of the learning ratemust be considered to fasten the learning speed anddecrease the error of training as much as possible.Thus η should tend to its maximum value, i.e. 1. Onthe other hand, when the Standard deviation E and itschange EC are both small, in this situation the NN sys-tem is close to a steady state, adjusting of the learningrate must be scaled down, it should tend to constantvalue or small value. According to the above anal-ysis, a set of fuzzy rules to adjust the learning rate ηcan be produced. Meanwhile, the improving measuressuch as additive momentum in virtue of determinationof momentum parameter reasonably has been taken,and better application result is acquired. The NN isthree layers feed-forward network. The number of in-put of NN is one; the number of hidden layer is 15nodes and the number of output is also one node. Thetimes of training are reached to 200. Certainly, thehigher accuracy of training could be attached by in-creasing the training epoch, but this will cost longercalculation time. The method proposed in this paperis implemented to compensate the measured stabilizedbiases for Gyro X , Y , Z at different temperatures re-spectively. The fuzzy control rules make the selectionof networks structure and learning rate η more rea-sonable. It exhibits a fast tuning process to reduce the

error of NN converging process.The measurement data are read into a MATLAB

program for preprocessing. Later F-NN calibrationprogram after training is stored in memory of FPGAnavigation system and applied online in experimentalsystem. In this way, the real-time of the FPGA navi-gation system is guaranteed in software.

The calculated values over the temperature range arechanged from -20◦C to 40◦C in steps of 5◦C. Eachfixed temperature point is kept for ten minutes. Atotal of 14 different temperatures are considered inthis experiment. The 3rd order polynomial thermalcompensation method [11] and combination of fuzzyand neural network(F-NN) thermal method proposedin this paper are both implemented to compensate themeasured biases for Gyro X at different temperaturesrespectively. The variations of stabilized biases withtemperature are shown in Figure 9.

Figure 9 Gyroscope biases with temperature

As observed from Figure 9, the biases of gyroscopesvary significantly with temperature. The maximumbiases can reach 20 deg/s over the entire temperaturerange. Hence there is a need for designing an accu-rate thermal calibration model for low cost MEMSsensors to compensate accurately for these bias driftswith temperatures. In Figure 9, the black dash-dottedline is the measured biases with temperature, whichis sampling data of output of IMU in calibration pro-cess. The red dash line is the compensation result with3rd order polynomial thermal compensation method.The black solid line is the compensation result with F-NN method. Comparing the three kinds of lines, wecan see that the thermal calibration method in F-NN isbetter than thermal method in 3rd order polynomial.

Considering IMU is used for attitude and positioncalculation process, the F-NN compensated methodcould eliminate the non-zero bias from raw data inMEMS IMU according to the feedback signals of tem-


perature sensor. The compensated precision in virtueof 3rd order polynomial method is within [-6, 6] deg/s.However, the precision with on the entire temperaturein F-NN thermal method is within [-0.5 0.5] deg/s.The above analysis indicates an almost perfect com-pensated result in elimination of more than 90% ofbias.

3.2 Combining with Kalman filter calibration me-thod

Dynamic Bias drift is a complex phenomenon whichis a combination of time, temperature and accelerationdependent behaviors. The dominant stabilized biasvalues are compensated in F-NN method with differ-ent temperatures in static test.

A Kalman filter is utilized to estimate the dynamicbias, random error as unknown rest calibration coeffi-cients. In order to implement Kalman filter in calibra-tion procedure, the simplified MEMS sensor Equation(5) are considered in calibration procedure. In mostcases, the misalignments are assumed small. The sim-plified formula cosθi j ≈ 1 is applied in (5). The termλ f i(i = x,y,z) is the 3-D vector of gyro scale factorerrors, δi j(i, j = x,y,z)is the 6-D vector of misalign-ment error, and δi j ≈ Si j(i, j = x, j,z) is misalignmentangles expressed in radians. Higher order indefinitesmall terms, such as λ f ·δ ≈ 0, d ≈ 0 are adopted, in(5). The misalignment (including cross-coupling) andscale factor error matrix N is given by

N =

λ f x δxz δxy

δyz λ f y δyx

δzy δzx λ f z

(13)

Therefore, the misalignment and scale factor matrixK is given by

K = 1+N =

1+λ f x δxz δxy

δyz 1+λ f y δyx

δzy δzx 1+λ f z

(14)

Taking MEMS gyro model as example, the mea-sured angular rate ωm in body frame in (5) is as fol-lows:

ωm = (I +N) ·ωt +br +β = ωt +∆ ·Γ+br +β (15)

where ωt is true angular rate in body frame, br is thebias drift of time varying component,β is the gyro biaswhich comprises the null shit bias b0 and the randomcomponent bg as the Gaussian White noise in Equa-tion (3).

∆ ·Γ = N ·ωt (16)

with

Γ =[

λ f x λ f y λ f z δxz δxy δyz δyx δzy δzx]T (17)

Γ =

ωtx 0 0 ωty ωtz 0 0 0 00 ωty 0 0 0 ωtx ωtz 0 00 0 ωtz 0 0 0 0 ωtx ωty

(18)

In Kalman filter, the 12-element state vector is de-fined as follows:

X = [Γ ξ ] (19)

where Γ is the misalignment and scale factor errorvector in (17), and ξ is the three-element bias driftof time varying component br in (15).

ξ = [brx bry brz]T (20)

The equation of state space process model becomes

X = AX +W (21)

where

A =[

0 diag(−1τ)3×3

]

12×12(22)

Reference [12] shows that temperature impact onscale factor is not important as on zero bias of gy-roscope. This is a likely model for the misalignmentparameters. Assuming that the scale factors are lit-tle affected by the expected temperature changes, asobserved by testing, this model should also be suffi-cient for the scale factor error parameters. The stabi-lized bias with fixed temperature is compensated byF-NN algorithm. However, the time-dependent gyrobias (gyro drift) is estimated in Kalman filter. The es-timated error would be subtracted from the raw dataof the gyroscope and accelerometer.

The measurement signal Z is defined to be the dif-ference between the commanded reference rate in ratetable and the measured rate of sensors, as follows:

Zk = ωm−ωt = HkXk +νk (23)

where the 3×12 measurement matrix H is given by

H = [I3×3 ∆]T (24)

The matrix ∆ is given by (18), and ν is the zero-meanGaussian white noise vector and null-shift bias com-pensated in F-NN. The Kalman filter is implementedin Figure 10.

The Kalman filter equations are used to determinethe estimate X and its covariance. The approximatemethod is applied in compensating the accelerometerin MEMS IMU.


Figure 10 Kalman filter algorithm

3.3 Experimental result and analysisCorresponding experiments are implemented in single-

axis rate table with thermal chamber. Concrete testingis executed as follows. Firstly define the fixed tem-perature in thermal chamber. Meanwhile, the com-manded rate in single axis rate table are given from -150◦/s to 150◦/s in steps of 10◦/s for 30 minutes. Laterthe temperature of the thermal chamber is varied tonext defined value. The full temperature is changedfrom -20◦C to 40◦C in steps of 5◦C. The same stepsare applied in rate table in reference to different tem-perature defined values. In this experiment, a total of13 different temperatures are considered and 31 dif-ferent rate data are obtained.

The experiment testing results of raw IMU error data,which comprise bias, misalignment and scale factorerrors are shown in variance rotating rates and tem-peratures are in Figure 11.

As observed from Figure 11, the raw IMU errors aremeasured in dynamic rate test and thermal test simul-taneously. It is evident to see that the raw errors ofgyroscopes vary significantly with variance tempera-tures and rotating rates. The maximum error can reach23 deg/s over the testing range for ADXR150 gyro Xin Figure 11 (a) and 300mg for ADXL210 accelerom-eter X in Figure 11 (b). Among these errors, the dom-inant error is the bias drift. Hence there is a need forimplementing an effective compensation method ofMEMS IMU accurately for these bias drifts. The posi-tion and attitude error in strapdown navigation systemwould occur without calibration and compensation ofIMU error.

Considering the variation of IMU error induced bydifferent dynamic range in rate test and temperaturevariation, the input of F-NN is two dimension vectors,which comprise the value of temperature of the ther-mal chamber and dynamic range of MEMS gyro oraccelerometer; the output of F-NN is the correspond-ing IMU bias drift in full temperature testing and dy-namic test. Similar adjusting principle is utilized in

(a)

(b)

Figure 11 Raw MEMS IMU error in real measurement

F-NN method. Rest small IMU errors are estimatedin Kalman filter. The compensated results using in-tegrated method combining F-NN and Kalman Filterare shown in Figure 12.

Comparing the result in Figures 11, 12, we can seethat the IMU error utilizing effective calibration andintegrated compensation approach is eliminated effec-tively. The compensated error of gyro is within [−0.050.05] deg/s in Figure 12 (a). The compensated meanerror of accelerometer is within 10mg deg/s in Figure12 (b). The experimental results indicate that the pro-posed integrated compensation method outperformsthe classical method. We guarantee a robust (againsttemperature variation and rate) sensor performance.

In the future, the fast and simplified algorithms ascalibration module of IMU would be developed to makethe feasibility further and high efficiency in navigationsystem based on FPGA and DSP. Moreover, furtherwork addresses the testing methodologies that havebeen used to effectively evaluate the continually evolv-ing MEMS based IMU over a variety of environments,including acceleration, vibration, shock, and tempera-ture. The effectiveness of the proposed integrated cal-ibration method is investigated further through a kine-matic van test using integrated GPS and compensated


(a)

(b)

Figure 12 Compensated MEMS IMU error

low-cost MEMS IMU.

4. Conclusions

This paper has presented a complete calibration pro-cedure for a low cost MEMS IMU with experimentalresults. A static, rate and thermal calibration proce-dure was performed using the raw output. The intelli-gent thermal-compensated method was integrated intoa Kalman Filter, which could not only eliminate domi-nant stabilized bias according to measurement of tem-perature sensor in static test, but also estimate randomerror, dynamic time varying bias, etc in rate test. Theprototype development board based on FPGA, dual-core processing could satisfy the demands of naviga-tion calculation in IMU in virtue of experimental test-ing. It costs 10∼ 15ms to accomplish the computationof INS navigation. Taking the advantage of proposedmethod, the MEMS IMU errors are compensated ef-fectively. The absolute rate error of the calibrated gyroin the selected application is within 0.04deg/s in entiredynamic thermal and rate range, and the absolute erroraccelerometer is within 10mg.

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