Time-asynchrony identi fi cation between inertial sensors in SIMU

Gongmin Yan,*,XiSun,Jun Weng,QiZhou2,and Yongyuan Qin

1.Schoolof Automation,Northwestern Polytechnical University,Xi’an 710072,China; 2.Xi’an Flight Automatic Control Research Institute,Xi’an 710065,China

Time-asynchrony identi fi cation between inertial sensors in SIMU

Gongmin Yan1,*,XiSun1,Jun Weng1,QiZhou2,and Yongyuan Qin1

1.Schoolof Automation,Northwestern Polytechnical University,Xi’an 710072,China; 2.Xi’an Flight Automatic Control Research Institute,Xi’an 710065,China

Traditionalstrapdown inertialnavigation system(SINS) algorithm studies are based on ideal measurements from gyros and accelerometers,while in the actual strapdown inertial measurement unit(SIMU),time-asynchrony between each inertial sensor is inevitable.Testing principles and methods for timeasynchrony parameter identi fi cation are studied.Under the singleaxis swaying environment,the relationships between the SINS platform drift rate and the gyro time-asynchrony are derived using the SINS attitude error equation.It is found that the gyro timeasynchrony error can be considered as a kind of pseudo-coning motion error caused by data processing.After gyro testing and synchronization,the single-axis tumble test method is introduced for the testing ofeach accelerometer time-asynchrony with respect to the ideal gyro triad.Accelerometer time-asynchrony parameter identi fi cation models are established using SINS speci fi c force equation.Finally,all of the relative time-asynchrony parameters between inertial sensors are well identi fi ed by using fi ber optic gyro SIMU as experimentalveri fi cation.

strapdown inertial navigation system(SINS),timeasynchrony,pseudo-coning error,velocity error.

1.Introduction

The traditional strapdown inertial navigation system (SINS)algorithms mainly focus on the compensation of attitude coning error,velocity sculling error and position scrolling error[1-7],where a basic assumption is implied thatallof the measurements from inertialsensors are fully synchronized in space and in time.However,for measurements from different sensors in the actual engineering strapdown inertialmeasurementunit(SIMU),there inevitably exist spatial and temporal asynchrony errors.In recent years,many effective methods have been acquired for spatial asynchrony issues[7-12],whereas the time asynchrony problem is comparatively concealed and complicated,still remaining as a hot topic to be resolved.Researchers have proposed a bunch of correction or compensation methods for coning and sculling errors based on the analysis of the frequency characteristics of actual inertial sensors[13-16].For example,in[15],in order to achieve time synchronization between accelerometer triad and gyro triad,accelerometer sampling signals are manipulated via a speci fi c digital fi lter,which is just the same fi lter used to process the high-frequency sampling signals forring laser gyros.However,the phase-frequency characteristics of these two types of inertial sensors,gyros and accelerometers,are mutually differentin actual SIMU,so the approach presented in[15]is quite incomplete.Reference[16]gives a correction method for sculling error compensation on the basis of accelerometerphase-shifterror,but itrelies on the precondition thatthe transfer function models of accelerometers have been obtained.Even so,these models are very dif fi cult to set up accurately in most cases for a real system.Besides,some other timedelay factors,like the signal transmission delays,are not comprehensively considered in these literature.

Based on the synchronized assumption between gyro triads and between accelerometer triads,experimental methods presented in[17]estimate the relative time-delay (or phase shift)between the accelerometer triad and the gyro triad.It has been proved that the time-delay effects are crucial in increasing the SINS velocity updating accuracy under the tumble test circumstance.However,if going a step further,itcan be concluded accordingly thatthe time asynchrony errors between gyro triads,and between accelerometer triads,have not yet been fully considered, which willseverely affecthigh-precision SINS,especially under a high-dynamic environment.These remaining issues are studied in depth in this paper,and testing methods are also broughtforward fortime-asynchrony between gyros,as well as between accelerometers.Finally,some testing experiments are carried outby using the fi ber-opticgyro(FOG)SIMUto realize time synchronization between allinertialsensors.

2.Principles for time-asynchrony testing between gyros

As shown in Fig.1,oxbybzbrepresents the body coordinate system of SIMU,denoted as b-frame for short.If SIMU experiences sinusoidal swaying about a fi xed axis oo＇with angular rateαΩcos(Ωt),it is obvious that the projections of the angular magnitude to SIMU coordinate axes areαiΩ(i=x,y,z)Therefore,the swaying angular velocity can be expressed in a componentform:

Fig.1 SIMU swaying along fixed axis oo＇

With the integral of(1)over time interval from 0 to t, the angular incrementis

Due to the swaying of fi xed-axis rotation in Fig.1,the corresponding equivalentrotation vector can be obviously simpli fi ed asΦ=θ.

Under a small swaying angle assumption and according to the relationship between the attitude transformation matrix and the equivalent rotation vector,an approximation could be given by

where I is a 3×3 identity matrix,Φ=|Φ|is the magnitude of equivalent rotation vectorΦ,anddenotes the transformation matrix from time t to 0.

The time-asynchrony effect of angular rate outputs between gyro triads should be considered in actual SIMU. Taking the navigation computer sampling time t as reference and assuming each time delay in the gyro triad to be τG,i(i=x,y,z),the actual outputs of the gyro angular velocity can be obtained by rewriting(1)as follows:

Subtracting(1)from(4),the gyro angular outputerrors can be obtained,as

If the‘East-North-Up’(E-N-U)local-level-northslaved coordinate system is selected as the navigation reference coordinate system,denoted as n-frame for short, the differential equation of SINS platform misalignment anglesφ=[φxφyφz]Tis given by[18]

After SINS initial fi ne alignmentand within a shortperiod of pure inertial navigation,the fi rst two terms in(6),i.e.are reasonably smallenough to neglect, so the following emphasis is puton the analysis of the last term.

Assume that b-frame is coincident with n-frame at the initialnavigation time 0.Replacingandδωrespectively in(6),and inserting with(3)and(5), yield

Expanding(7)and analyzing the x-component,it is found that

Assuming a small product valueΩτG,ibetween the swaying frequency and the time-asynchrony delay,there is the approximation

Substituting(9)into(8),the DC componentin˙φxcan be given by

Similarly,the DC components of˙φyand˙φzin(7)are respectively shown as

Itis wellknown thatthere existnoncommutativity errors in SINS attitude updating for fi nite rotation,which means the SINS attitude updating is highly affected by the rotation orderofangularmovements.Underan angularmotion environment,the time-asynchrony outputs from gyros will result in a false attitude updating,which is not well consistent with actual angular motion sequence and will lead to attitude noncommutativity errors.Therefore,the timeasynchrony error can be essentially seen as a kind of noncommutativity errors caused by attitude computation,or it can be regarded as a pseudo-coning motion error[19,20]. Thus,in this paper,this error is called time-asynchrony pseudo-coning error due to phase-shiftintroduced by nonidealgyro outputs.

In particular,assumeα=10?,Ω=2πrad/s(or 1 Hz) andτG,x-τG,y=0.01 ms in(13),then an average misalignment drift rate of˙ˉφz=0.62(?)/h can be obtained, which is much larger than the drift error of the inertialgrade gyro(0.01(?)/h).This numericalexample shows that much more attention should be paid to high precision SINS especially underan intense circumstance.

Conversely,the above analyses also provide a perfect idea for the time-asynchrony error identi fi cation between gyros,which can be implemented by introducing timeasynchrony pseudo-coning errors through experimental methods.If SIMU undergoes fi xed-axis swaying motion using the turntable under the laboratory condition,then with no much arrangement from(10)-(12),the following matrix equation can be easily gotten where sinusoidalswaying frequencyΩ,amplitudeαi,and misalignmentangle drift rateare all known parameters in the experiment.

Unfortunately,in(14)the rank of the coef fi cientmatrix aboutthe unknown time-asynchrony parameters is 2.As a result,only the relative time-asynchrony,τG,i-τG,j(i,j= x,y,z;i/=j),between any two gyros will be calculated, but none of the absolute time-asynchronyτG,ican be separately identi fi ed.

When obtaining the time-asynchrony parameters between gyro triads,a linear extrapolation or interpolation algorithm can be used to exactly adjustthe outputtimes of any two gyros to the third gyro,so that all of the output times in gyro triads will be well synchronized.The synchronization algorithms were derived in[17].In the following section for accelerometer time-asynchrony analysis,we assume thatthe time-asynchrony between gyros has been fully tested and compensated,thatis to say,the gyro outputs are considered to be ideally synchronized.

3.Principles for time-asynchrony testing between accelerometers

As shown in Fig.2,the SIMU oybaxis lies in horizontal and north directions,pointing inside and being perpendicularto the paperplane in this diagram.If b-frame coincides with n-frame atthe initialnavigation time 0,and SIMU rotates around the oybaxis with a constantangular rateωyto rollangleγ,then oxbzbrepresents the new orientation of b-frame computed and determined by the ideal gyro triad. Thus,the SIMUattitude direction cosine matrix can be described as

Fig.2 Time-delay diagram for accelerometer rolling motion

Atthe angularposition oxbzb,the idealspeci fi c force,or named as virtual speci fi c force,projecting in b-frame can be calculated via transformation matrix Cnb,with the result beingwhere gn=[0 0-g]Tand g is the magnitude of local gravity,which is approximately 9.8 m/s2.

However,because ofthe time-asynchrony ofaccelerometeroutputs with respectto gyro triads,the sensitive axis of the oxb-axis accelerometeris actually along the oxAorientation,as shown in Fig.2.Hence,there exists a small roll angle errorδγx=∠x(chóng)Aoxb=γ-?γx,and then the actually sensed output of the oxb-axis accelerometer is expressed by

where sinδγx≈δγxand cosδγx≈1.

Similarly,the actual sensitive axis of the ozb-axis accelerometer is along the ozAorientation.By de fi ning anotherrollangle errorδγz=∠zAozb=γ-?γz,the ozb-axis accelerometeroutputis given by

Whereas,for the oyb-axis accelerometer,its sensitive axis is always along the horizontaldirection with zero output,?fbsf,y≡0,so the oyb-axis accelerometer is absolutely insensitive to the time-asynchrony in the oyb-axis tumble test.

In the following partofthis section,two data processing methods for deriving accelerometer time-asynchrony parameters willbe introduced.For convenience and simpli fication,the symbolτA,iis denoted as the time-asynchrony of the oib-axis(i=x,y,z)accelerometer with respect to gyro triads.

3.1 Method I

By combining actual speci fi c force outputfrom the oxb-axis accelerometer with the virtual speci fi c forcesfrom the other two axes,here a new speci fi c force vector is de fi ned as

Let roll angleγ=ωyt and the roll angle error of the oxb-axis accelerometerδγx=ωyτA,x.After m cycles of uniform rolling,i.e.a rotation of m×360?,the variation of the SINS eastern velocity can be obtained from the integral of(20),with the following result

Equations(21)and(22)show thatif SIMU rotates about an axis along the horizontaland north orientations,the accelerometer time-asynchrony parameters of the other two axes can be identi fi ed from the SINS eastern velocity errors by constructing new speci fi c force vectors and implementing SINS velocity updating.

3.2 Method II

In contrastwith Method I,directly applying the actualspec i fi c force output vector

Similar to(21),assuming m cycles of uniform rolling rotation and integrating(23),the variation of SINS eastern velocity errors is given by

For example,if m=1 andτA,x=1 ms,the velocity variation is approximately

On the other hand,if the new speci fi c force vector is constructed as,by using the same data processing method shown in(20)and(21),it can be easily obtained thatObviously,parametersτA,xandτA,zcannot be separated in(24)by only one single-axis rotation testing.Assuming τA,x=τA,z,(24)is completely consistentwith the result in[17].Therefore,the research in[17]can be seen as a specialcase study in this paper.

Itis noted thatthere are some differences between these two data processing methods.In Method II,actualspeci fi c force outputs are directly used for the navigation velocity updating.It avoids the inconvenience of constructing virtualspeci fi c forces,butthree rotation tests,by placing oxb, oyband ozbaxes in the horizontal and the north orientations respectively,are demanded to separately identify all ofthe time-asynchrony parameters.While in Method I,two time-asynchrony parameters can be obtained within the individual rotation test.Therefore,only two rotation tests, such as rotations about oxb-axis and oyb-axis,are enough to solve allof the three parameters.

4.Testing and analyses

The fi ber optic gyro(FOG)SIMU and the swaying turntable are two kinds of main equipmentused in the following laboratory experiments,where the FOG-SIMU is close to inertial-grade with a gyro bias stability of about 0.01(?)/h and an accelerometer bias repeatability of about 5×10?5g.Although the gyro and accelerometer outputs for this FOG-SIMU are in angular and velocity increment mode,it can be proved easily that the results from(14), (22)and(24)are stillapplicable.

The main performances of the turntable are as follows: angular position accuracy 2＇,angular rate accuracy and smoothness 5×10?5(within 360?),swaying amplitude≥10?,swaying frequency≤1 Hz.

4.1 Gyro testing

Firstly,the FOG-SIMU is installed on the swaying turntable with xbybzbcoordinate axes approximately pointing to E-N-U geographic orientations respectively. Aftera 10 min initial fi ne alignmenton the stationary base, FOG-SIMU goes into a navigation mode of attitude updating,and the velocity updating can be negligible ordirectly setto 0.Record the turntable’s initialangular position and keep stationary for a shorttime,then undertake sinusoidal swaying about the diagonal of FOG-SIMU oxband oybaxes with swaying amplitude 10?and frequency 1 Hz.The swaying duration is about33 s.When it stops,restore the turntable to its initialposition and keep stationary forshort time.

Fig.3 shows the azimuth variation in the SINS attitude updating process.It indicates that azimuth variation changes very slowly in both stationary stages,butan obvious and linear drifttrend occurs during the swaying stage, which is obviously caused by the time-asynchrony error between oxband oybgyros.

Fig.3 Relative azimuth variation for gyro testing

In Fig.3,the azimuth variation between two stationary stages is about-0.016?,which is equivalent to the misalignment drift angleSubstituting the aforementioned experimental parameters into(13),yields then gives the solution with arrangementτG,x-τG,y= -0.028 ms.

Due to the gyro random drift error,about 0.001?variation in Fig.3 during stationary stages,plus the turntable accuracy 2＇＇,the uncertainty of the gyro time-asynchrony error is estimated by 0.028 ms×(0.001?+2＇＇)/0.016?≈0.003 ms.

Similarly,when the turntable sways about the diagonal of FOG-SIMU ybzbaxes,the resultofτG,y-τG,z= 0.039 ms can be obtained from the testing.While swaying aboutthe diagonalof SIMU ybzbaxes,τG,z-τG,x= -0.017 ms is gotten.

Finally,it is easy to verify that the above three timeasynchrony error values in these tests can almost form a closed-loop relationship,as(τG,x-τG,y)+(τG,y-τG,z)+ (τG,z-τG,x)=(-0.028 ms)+0.039 ms+(-0.017 ms)= -0.006 ms≈0.

4.2 Accelerometer testing

In the previous subsection ofgyro time-asynchrony testing and compensation,the ozb-axis gyro output time is taken as synchronization reference,so the gyro triad is considered to be well synchronized.In this subsection,assume that inner lever arm errors for accelerometers have been pre-compensated before the time-asynchrony testing.

Similar to the FOG-SIMU installation for gyro testing,let xbybzbcoordinate axes approximately point to E-N-U geographic orientations respectively.After a 10 min initial fi ne alignmenton the stationary base,FOGSIMU carries out both attitude and velocity updating simultaneously.In the attitude updating algorithm,it takesadvantage of the gyro data with time-asynchrony compensation to minimize attitude errors.While in the velocity updating algorithm,it utilizes the new speci fi c force vector constructed by(19)as the input.Rotate the turntable about FOG-SIMU oyb-axis with an angular rate 50(?)/s. After each cycle of turntable rotations,keep it stationary for a short time,and a total rotation of three cycles is experienced.Using the data processing method presented in Subsection 3.1,the SINS eastern velocity variation is obtained and shown in Fig.4.

Fig.4 Eastern velocity variation for accelerometer testing

In Fig.4,the totaleastern velocity variation within three cycles of turntable rotation is 0.262 m/s.Substituting the above experimental parameters into(21),we will get the solution.

In a similar way as gettingτA,x,we will haveτA,y= 2.262 ms andτA,z=2.442 ms according to more laboratory experiments,butwith no detailnecessarily repeated here.

To sum up allthe testing results ofgyros and accelerometers,a schematic diagram for time-asynchrony relationships between FOG-SIMU sensors is shown in Fig.5, where Gx,Gy,Gz,Ax,Ay,Azare denoted as gyros and accelerometers respectively,

Fig.5 Time-asynchrony diagram for FOG-SIMU

It shows that the time-asynchrony errors between gyros are relatively small,only in the order of 0.01 ms,and the time-asynchrony errors between accelerometers are in the order of 0.1 ms,while the time-asynchrony errors between gyros and accelerometers are comparatively large, in the order of 1 ms.Note that the time interval between the gyro Gzoutputand the navigation computer sampling time tihas not yet been solved by the testing method in this paper.Surely,this time difference affects little on the accuracy of the pure inertial navigation algorithm,but it is likely to bring negative impactwhen SIMU is integrated with otherdevices,such as in high-precision SINS/GPS integrated navigation solution,which needs further studies.

5.Conclusions

In actual SIMU,in addition to the time-asynchrony between the gyro triad and the accelerometer triad,the timedelay characteristics of each gyro or each accelerometer are always notidentical,and these will also lead to corresponding time-asynchrony errors in SINS attitude and velocity updating algorithms.In this paper,the formulas between time-asynchrony parameters and platform misalignmentangles,and between time-asynchrony parameters and navigation velocity errors,are derived in detail,which provides a theoretical basis for laboratory testing design and also provides methods for time-asynchrony parameter identi fi cation.Finally,relative time-asynchrony parameters between inertialsensors are obtained in experimental testing using FOG-SIMU.The testing results show thatthe time-asynchrony parameters between gyros or between accelerometers are much smaller than thatbetween the gyro triad and the accelerometertriad.Even so,more attention is worthy to be paid to these small parameters especially for high-precision SINS under the severe environment.Carefultesting and compensation is of greatimportance for the improvementof SINS navigation accuracy.

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Biographies

Gongmin Yan was born in 1977.He received his B.S.degree in automatic controland Ph.D.degree in navigation,guidance and control in Northwestern Polytechnical University in 2000 and 2006,respectively.Now he is an associate professor in the School of Automation,Northwestern Polytechnical University.He has authored more than 30 scienti fi c publications including one book.His research interests include inertial navigation,integrated navigation and data fusion.

E-mail:yangongmin@163.com

Xi Sun was born in 1990.She received her B.S. degree in automation in Northwestern Polytechnical University in 2012.Now she is a postgraduate student in the School of Automation,Northwestern Polytechnical University.Her research interests are inertialnavigation and IMU calibration.

E-mail:susanxi2008@163.com

Jun Weng was born in 1988.He received his M.S.degree in precise instrument and mechanics from Northwestern Polytechnical University in 2013.Now he is a Ph.D.candidate in the School of Automation,Northwestern Polytechnical University.His research interests mainly include inertial navigation and integrated navigation.

E-mail:npu wengjun@sina.com

QiZhou was born in 1984.He received his B.S.and Ph.D.degrees in Northwestern Polytechnical University in 2007 and 2013,respectively.Now he is an engineer in Xi’an Flight Automatic Control Research Institute,Aviation Industry Corporation of China.His research interests are inertial navigation and integrated navigation.

E-mail:zhouqis@139.com

Yongyuan Qin was born in 1946.He received his M.S.degree of engineering in Northwestern Ploytechnical University in 1981.He is a professor and a Ph.D.supervisor in the Schoolof Automation, Northwestern Polytechnical University.His research interests include gyro technology,inertial navigation,Kalman fi ltering,data fusion and fault detection.

E-mail:qinyongyuan@nwpu.edu.cn

10.1109/JSEE.2015.00040

Manuscriptreceived April18,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61273333).