Research ArticleAnalysis and Improvement of Attitude Output Accuracy inRotation Inertial Navigation System

Kui Li12 Pengyu Gao2 Lei Wang2 and Qian Zhang2

1School of Electronic and Information Engineering Beihang University Beijing 100191 China2School of Instrument Science and Opto-Electronics Engineering Beihang University Beijing 100191 China

Correspondence should be addressed to Kui Li xflikui126com

Received 7 May 2015 Accepted 2 July 2015

Academic Editor Jian Guo Zhou

Copyright copy 2015 Kui Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Inertial navigation system (INS) measures vehiclersquos angular rate and acceleration by orthogonally mounted tri-axis gyroscopes andaccelerometers and then calculates the vehiclersquos real-time attitude velocity and position Gyroscope drifts and accelerometer biasesare the key factors that affect the navigation accuracy Theoretical analysis and experimental results show that the influence ofgyroscope drifts and accelerometer biases can be restrained greatly in rotation INS (RINS) by driving the inertial measurementunit (IMU) rotating regularly thus improving navigation accuracy significantly High accuracy in position and velocity should bematched with that in attitude theoretically since INS is based on dead reckoning However the marine and vehicle experimentsshow that short-term attitude output accuracy of RINS is even worse compared with that of nonrotation INS The loss of attitudeaccuracy has serious impacts on many task systems where high attitude accuracy is required This paper researched the principleof attitude output accuracy loss in RINS and then proposed a new attitude output accuracy improvement algorithm for RINSExperiment results show that the proposed attitude compensation method can improve short-term pitch and roll output accuracyfrom 20sim30 arc seconds to less than 5 arc seconds and azimuth output accuracy improved from 2sim3 arc minutes to less than 05 arcminutes in RINS

1 Introduction

Inertial navigation systems (INS) employ inertial sensors(gyroscopes and accelerometers) to establish an inertialplatform for measuring vehiclersquos angular rate and accel-eration and then calculate vehiclersquos attitude velocity andposition based on dead reckoning principle Traditionalinertial platforms are usually constructed physically by threegimbals at least to isolate vehiclersquos angular motion With thedevelopment of technology inertial platform INS is replacedby strapdown INS gradually in which the inertial sensors arefixed along body axes and the physical platform is constructedmathematically by the attitude transformation matrix Strap-down configurations could reduce system cost and increasesystem reliability greatly compared with physical platformINS while in both physical platform system and strapdownsystem position errors accumulate with time because ofgyroscope drifts and accelerometer biases However if theinertial measurement unit (IMU) is forced to rotate along

given axes regularly gyroscopes drifts and accelerometer biaserrors can be modulated from constant to periodically vary-ing components thus attenuating system errors prominently

Research on rotation INS (RINS) has been carried out inthe past Pioneering research was presented by Geller in 1968which described an inertial platform INS with continuousplatform rotation relative to the local geodetic frame [1]Laser gyroscopes applied to RINS emerged from the 1980s[2 3] A laser gyroscope RINS with discontinuous rotationwas proposed by San Giovanni Jr and Levinson in 1981 [4]In 2004 Yang and Miao analyzed single-axis continuousrotation INS based on fiber-optic gyroscopes [5] In 2007Ishibashi et al proved that position accuracy would acquiregreat improvement by driving INS rotate continuously ona turntable during alignment and navigation [6 7] Dual-axis RINS was brought out to solve the defects that verticalaxis inertial sensors errors cannot be restrained in single-axis RINS [8 9] In dual-axis RINS whether it is based oncontinuous rotation scheme [10] or consequential rotation

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 768174 10 pageshttpdxdoiorg1011552015768174

2 Mathematical Problems in Engineering

Encoder

Torque motor

Az

Gz

Gy AyGx

Ax

Oys

xs

Figure 1 System structure

scheme [11] inertial sensor errors along three directions canbe restricted simultaneously Besides the USA has startedthe IFOG strategic nuclear submarine plan and expected tomanufacture the first tri-axis RINS system in 2010 [12] Thetri-axis RINS system is designed with three-axis continuousrotation scheme [13] and can compensate the scale factorerror of sensors installation axis instability gyroscope driftsand so forth These papers demonstrate that RINS has beenwidely researched and has huge prospect for improving INSnavigation accuracy However most research on RINS aimsat velocity and position accuracy improvement which isalso verified by RINS experimental results in this paperBut it should be noticed that RINS experimental resultsalso show that short-term attitude output accuracy of RINSgets even worse compared with that of nonrotation INSIn some airborne shipborne or land vehicles applicationshigh accuracy attitude information output is more crucial forsome task systems and loss of attitude accuracy will lead toserious consequences Since INS is based on dead reckoningprinciple high accuracy in position and velocity should bematched with attitude accuracy This paper researched thereasons that caused the loss of RINS attitude output accuracyand then presented the corresponding attitude accuracyimprovement algorithm Section 2 described the RINS sys-tem configuration and analyzed errors restraint principle andpreliminary experiment results Section 3 presented RINSattitude update algorithm and researched the possible rea-sons of attitude output accuracy loss Section 4 proposedthe designed RINS attitude compensation algorithm andpresented the experimental verification results Conclusionswere drawn in Section 5

2 RINS Error Restraint Analysis andExperiment Verification

21 RINS Error Restraint Analysis The system structure ofthe experimental RINS is shown in Figure 1The IMUmainlyincludes three Ring LaserGyroscopes (RLG) and three quartzaccelerometersThe torque motor drives IMU to rotate alongthe vertical body axis bidirectionally and an angular encoderis fixed to provide rotation angle relative to the body frame

Table 1 Main componentsrsquo specification of RINS

Gyro drifts 0008∘hAccelerometers biases 50 120583gEncoder resolution 1081015840pulseRotation angular rate 0686∘s

O

ys

120593

120593

xbxs

yb

zb(zs)

Figure 2 Frame definition

Table 1 shows the main componentsrsquo specification of theexperimental RINS

The proposed encoder provides digital output of 20000pulses per 360∘ rotation and the resolution is 1081015840pulseThe experimental RINS takes the scheme of bidirectionallycontinuous rotation and the rotation period is 1050 s Con-sidering IMU rotation the experimental RINS involves a newframe named inertial sensing frame (119904 frame O-xsyszs) Asis shown in Figure 2 119904 frame refers to the rotation framewhich varies with the change of inertial sensorsrsquo pointingdirection in real time The body frame (119887 frame O-xbybzbright-forward-upward) is defined the same as 119904 frame whenrotation angle 120593 is zero

The errors restraint principle of RINS can be explained bythe following analysis At time 119905 the angular rate and specificforce output of the gyroscopes and accelerometers are

120596

119904

119894119904

= 119862

119904

119887

120596

119887

119894119887

+ [120576

119909120576

119910120576

119911]

119879

+120596

119904

119887119904

119891

119904

119894119904

= 119862

119904

119887

119891

119887

119894119887

+ [nabla

119909nabla

119910nabla

119911]

119879

(1)

where 120596119887119894119887

and 119891

119887

119894119887

denote ideal gyroscopes and accelerom-eters outputs along 119887 frame 119862119887

119904

denotes the coordinate-transformationmatrix between 119887 frame and 119904 frame 120576

119909 120576119910 120576119911

are gyroscope drifts and nabla119909 nabla119910 nabla119911are accelerometer biases

120596

119904

119887119904

= [0 0 120596]

119879 represents rotation rate projected on 119904 frameAfterwards the angular rate and the specific force in body

frame can be obtained by the following equations

119887

119894119887

= 119862

119887

119904

120596

119904

119894119904

+120596

119887

119904119887

119891

119887

119894119887

= 119862

119887

119904

119891

119904

119894119904

(2)

Mathematical Problems in Engineering 3

where

119887

119894119887

and

119891

119887

119894119887

represent gyroscope and accelerometeractual outputs and 120596

119887

119904119887

= minus119862

119887

119904

120596

119904

119887119904

= [0 0 minus120596]

119879 Then (3)can be obtained by substituting (1) into (2) as follows

[

[

[

[

119887

119894119887119909

119887

119894119887119910

119887

119894119887119911

]

]

]

]

= 120596

119887

119894119887

+

[

[

[

120576

119909cos120593 minus 120576

119910sin120593

120576

119909sin120593 + 120576

119910cos120593

120576

119911

]

]

]

[

[

[

[

119891

119887

119894119887119909

119891

119887

119894119887119910

119891

119887

119894119887119911

]

]

]

]

= 119891

119887

119894119887

+

[

[

[

nabla

119909cos120593 minus nabla

119910sin120593

nabla

119909sin120593 + nabla

119910cos120593

nabla

119911

]

]

]

(3)

where 120593 denotes the rotation angle between 119887 frame and 119904

frame provided by the encoder If the rotation rate of IMU isideal the rotation angle 120593 equals 120596119905 during forward rotationand minus120596119905 during reverse rotation

In traditional strapdown INS constant inertial sensorerrors (gyro drifts and accelerometer biases) in 119887 frame arethe main contributor to INS positioning error while in RINSinertial sensor errors turn into periodical components Thenafter dead reckoning calculation attitude errors horizontalvelocity errors and horizontal position errors caused bygyroscope drifts and accelerometer biases can be modulatedso navigation errors in RINS are no longer accumulated withtime and navigation accuracy could achieve great improve-ment

22 RINS Experimental Verification Results In order tocompare the error restraint effect brought about by IMUrotation the experimental RINS can be transformed intotraditional strapdown INS by locking the torque motorrotation axis while navigation algorithm remains unchangedUnder the sameworking condition comparative experimentsare conducted in both rotation and strapdown mode whichlast 4200 s (4 bidirectional rotation periods) The proposedRINS prototype and associated experimental equipment andtheir connections are shown in Figure 3The RINS prototypeis mounted on a dual-axis turntable the experimental dataare collected by a laptop

Experimental results are given as follows Figure 4 showshorizontal position errors Figure 5 shows horizontal velocityerrors and Figure 6 shows attitude errors The dashed curverepresents strapdown results and the solid curve representsrotation results

In Figures 4 and 5 E is short for east and N is shortfor north The maximum of horizontal positioning errorin the strapdown mode is 1500m during experiment whilemaximum positioning error is only 160m in rotation modewhich means that positioning accuracy is improved nearly10 times in RINS The maximum of horizontal velocity errorin the strapdown mode is 15ms while maximum velocityerror in rotation mode is less than 03ms It means velocityaccuracy is improved about 5 timesThus it can be seen thatby driving IMU rotation bidirectionally horizontal positionand velocity accuracy are improved obviously in RINS

The attitude outputs during comparative experiment aregiven in Figure 6 It can be seen that in strapdown mode

Turntable

Power supply

RINS

Laptop

Figure 3 RINS and associated experimental equipment

attitude error changes slowly during experiment and it showsgood stability in short term (several minutes) The relativelylong-term periodical varying components in pitch and rollerrors are Schuler oscillations (844min period) Howeverin rotation mode attitude errors become more complicatedPitch and roll errors are stable on the general trend duringthe whole 70-minute experiment The mean value of attitudeerrors in every rotation period is nearly the same It is hardto see Schuler oscillations in the error curve because mostof horizontal inertial sensor errors have been attenuatedby IMU rotation thus Schuler oscillations amplitude issmall However the short-term periodical fluctuation arisesin attitude outputs with oscillations amplitude of 20sim30 arcseconds in pitch and roll errors and 2sim3 arc minutes inazimuth errors Compared to strapdown mode position andvelocity accuracy inRINS increased greatly but attitude errorsfail to reach the same accuracy in short term (even getworse) let alone improvement by IMU rotation But for someairborne shipborne or land vehicle task systems not onlyis high accuracy positioning and velocity reference neededbut also high accuracy attitude reference is important Errorfluctuation of 20 arc seconds in pitch and roll output anderror fluctuation of 2 arc minutes in azimuth output willhave seriously negative effect on their performance Howeveraccording to theoretical analysismentioned in Section 2 highaccuracy in position and velocity should be matched withhigh attitude accuracy since INS is based on dead reckoningprinciple Therefore it is very necessary to analyze causes ofthe attitude output fluctuation errors and to find solutions toimprove short-term attitude accuracy in RINS

3 Analysis of Attitude OutputAccuracy in RINS

31 AttitudeUpdate Algorithm in RINS Thebodyrsquos attitude inINS can be described by transformation matrix from 119887 frameto 119899 frame (navigation frame) through attitude matrix 119862119899

119887

In

4 Mathematical Problems in Engineering

minus2000

minus1000

0

1000

Time (s)

PE er

ror (

m)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 40000 500 1000 1500 2000 2500 3000 3500 4000minus500

0

500

1000

Time (s)

PN er

ror (

m)

RotationStrapdown

Figure 4 PE and PN errors comparison

0 500 1000 1500 2000 2500 3000 3500 4000minus2

minus1

0

1

Time (s)

VE

erro

r (m

s)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 4000minus05

0

05

1

Time (s)

VN

erro

r (m

s)

RotationStrapdown

Figure 5 VE and VN errors comparison

0 500 1000 1500 2000 2500 3000 3500 4000minus50

0

50

Time (s)

RotationStrapdown

Pitc

h er

ror (

998400998400)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 4000minus50

0

50

Time (s)

Roll

erro

r (998400998400

)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 4000minus200

0

200

Time (s)

Azi

mut

h er

ror (

998400998400)

Figure 6 Strapdown and rotation INS attitude errors comparison

strapdown INS configuration 119862119899119887

is calculated directly fromrotation vector by quaternion attitude updating algorithmHowever in RINS configuration attitude matrix 119862119899

119904

shouldbe updated beforehand (the update process ofmatrix119862119899

119904

is thesame as that of119862119899

119887

in strapdown INS) then119862119899119887

can be acquiredaccording to the following equation

119862

119899

119887

= 119862

119899

119904

119862

119904

119887

(4)

The matrix 119862119904119887

is the transformation matrix from 119887 frameto 119904 frame and is given as follows

119862

119904

119887

=

[

[

[

cos120593 sin120593 0minus sin120593 cos120593 0

0 0 1

]

]

]

(5)

where 120593 is the rotation angle provided by encoder

Mathematical Problems in Engineering 5

If all the updating processes above are ideal attitudeoutput in RINS should present high accuracy and shouldnot contain short-term error fluctuation As a matter of factin RINS updating matrix 119862

119899

119904

is calculated from rotationvector by quaternion attitude updating algorithm which isthe same as 119862119899

119887

in strapdown INS thus it is accurate enoughConsequently the most possible cause of RINS short-termattitude output accuracy loss is transformation process fromattitude matrix 119862119899

119904

to 119862119899119887

that is to say matrix 119862119904119887

In actualRINS the encoder output errors installation eccentricity ofencoder and noncoaxial rotation of IMU can cause rotationangle 120593 containing periodical output angle errors [14] whichlead to azimuth output fluctuation inRINS Besides if the axisO-zs does not coincidewith the rotation axisO-zb all the timethe rotation angular rate will have projection along horizontalaxes O-xs and O-ys Thus undesirable influences will beattached to the transformation matrix 119862119904

119887

which will resultin error fluctuation on pitch and roll output The detailedanalysis will be given as follows

32 Analysis of Azimuth Output Error in RINS Equation (5)indicates that the rotation angle 120593 will have effect on theaccuracy of matrix 119862119904

119887

Therefore it is necessary to analyzethe rotation angle provided by the encoder first In orderto obtain the encoder angle errors the angle integrated byoutput angular rate of gyroscope 119885 is used However outputof gyroscope 119885 contains not only rotation angular rate 120596 butalso gyroscope drift 120576

119911and the upward component of earth

rotation angular rate 120596119894119890sin 119871 The sign of IMU rotation rate

120596 is opposite during forward and reverse rotation Then thegyroscope outputs are given as

120596

119911119891= 120596+120596

119894119890sin 119871+ 120576

119911

120596

119911119903= minus120596+120596

119894119890sin 119871+ 120576

119911

(6)

where subscript 119891 denotes forward rotation and 119903 denotesreverse rotation If we define 1205961015840 as 120596

119894119890sin 119871 + 120576

119911= (120596

119911119891+

120596

119911119903)2 then the angle integrated by output angular rate

of gyroscope 119885 during IMU rotation can be calculatedaccording to (7) which will be a reference for encoder angleoutput Consider

120593

119911119891= int (120596

119911119891minus120596

1015840

) 119889119905

120593

119911119903= int (120596

119911119903minus120596

1015840

) 119889119905

(7)

Figure 7 shows actual experimental data of encoder angleoutputs and Figure 8 is acquired when subtracting encoderangle from the integrated angle of gyroscope 119885 It can beseen from the two figures that encoder angle error (defineas 120575120593) presents obvious repeatability during 4 bidirectionalrotation periods and the fluctuations are axial symmetryabout forward and reverse rotation in a separate bidirectionalperiod 120575120593 presents periodical fluctuation with the sameperiod as IMU rotation and the fluctuation amplitude is 1sim2 arc minutes It presents the same property as azimuth errorin Figure 6 and can be compensated afterwards short-termazimuth output accuracy would be greatly improved

0 500 1000 1500 2000 2500 3000 3500 40000

50

100

150

200

250

300

350

400

Time (s)

Enco

der o

utpu

t (∘ )

Figure 7 Encoder angular output

0 500 1000 1500 2000 2500 3000 3500 4000minus150

minus100

minus50

0

50

100

150

Time (s)

Enco

der a

ngle

erro

r (998400998400

)

Figure 8 Encoder angular error

33 Analysis of Pitch and Roll Output Error in RINS Theexpression of matrix 119862

119904

119887

in (5) is obtained based on idealrotation namely the axes of O-xs and O-ys stay in a fixedplane during rotation However in fact irregular rotation ishard to avoid and the rotation planewould change in differentpositions The difference between actual rotation and idealrotation should be analyzed in detail

Figure 9 (left) describes IMU rotation under ideal condi-tion and in this case axes of O-xs and O-ys in 119904 frame remainunchanged in fixed plane and axis O-zs coincides with O-zbin body frame all the time However in actual condition dueto some reasons such as machining and assembling errors inrotation axis and defects in rotation bearings the IMU rotatesirregularly Figure 9 (right) shows this irregular rotation andthe rotation plane in 119904 frame no longer stays unchanged thepointing of O-zs axis will change in space and axes of O-xsand O-ys will fluctuate around the ideal fixed plane

If the rotation plane remains unchanged the projectionof acceleration of gravity on horizontal accelerometers will

6 Mathematical Problems in Engineering

Actual rotation axis

Rotation axis

ys

Ay

Az

xs

xs

AxAx

zszb

ysAz

Ay 120575120579

120575120574

zb(zs)

Figure 9 Comparison of ideal rotation and actual irregular rotation

0 500 1000 1500 2000 2500 3000 3500 4000minus004

minus002

0

002

004

Time (s)0 500 1000 1500 2000 2500 3000 3500 4000

minus004

minus002

0

002

004

Time (s)

Acce

lero

met

er x

(ms2)

Acce

lero

met

er y

(ms2)

Figure 10 Initial outputs of horizontal accelerometers

0 500 1000 1500 2000 2500 3000 3500 4000minus200

minus100

0

100

200

Time (s)0 500 1000 1500 2000 2500 3000 3500 4000

minus200

minus100

0

100

200

Time (s)

Resid

ual o

f acc

elero

met

er120583

g)

Resid

ual o

f acc

eler

omet

er120583

g)

x (

y (

Figure 11 Residual components of horizontal accelerometers

be standard sine or cosine form with the same period asIMU rotation and the amplitude is determined by initialpitch and roll of the rotation plane However due to thefluctuation of rotation plane in actual situation the projectionof acceleration of gravity on horizontal accelerometers willbring about irregular variation and the deviation betweenthe rotation plane and the fixed plane can be representedby irregular variation in horizontal accelerometers Figure 10shows the original output of two horizontal accelerometers inactual experiments In order to obtain the irregular rotationcomponents firstly the projection of acceleration of gravitycaused by pitch and roll should be deducted from thehorizontal accelerometers according to

119886

1015840

119909

= 119886

119909+1198921205740 cos120593minus1198921205790 sin120593

119886

1015840

119910

= 119886

119910minus1198921205790 cos120593minus1198921205740 sin120593

(8)

where 1205790 1205740 are initial pitch and roll acquired by thealignment process in RINS

Figure 11 shows the residual components of horizon-tal accelerometers during the experiment Both horizon-tal accelerometers residual components mainly representsecond harmonic frequency related to the rotation periodand show good repeatability during 4 bidirectional rotationperiods In a single bidirectional period the fluctuationson the horizontal accelerometers are also axial symmetryThe fluctuation amplitude is about 150 120583g which means themaximum deviation between the rotation plane and the fixedplane is about 30 arc seconds and it coincides with theamplitude of pitch and roll output fluctuation in Figure 6

The residuals of horizontal accelerometers can be trans-formed into deviation angles between the rotation plane and

Mathematical Problems in Engineering 7

120575120574

120575120574

120575120579

120575120579O

zs

xs

ys

y998400s

x998400s

z998400s

Figure 12 Transformation relation between s frame and 1199041015840 frame

the fixed plane which are defined as 120575120579 120575120574 as shown inFigure 9 (right) and are given as follows

120575120579 =

119886

1015840

119910

119892

120575120574 = minus

119886

1015840

119909

119892

(9)

To compensate influence caused by 120575120579 120575120574 anothercoordinate frame should be defined in the proposed RINS todistinguish actual irregular rotation from ideal rotation Theideal rotation frame is named as 1199041015840 frame whose O-1199111015840

119904

axis iscoinciding with O-zb axis all the time The previous 119904 frameis the actual rotation frame representing the instantaneousrotation frame and it fluctuates around the ideal fixed planeTaking into consideration irregular rotation attitude matrixobtained by (6) should be described as 119862119899

1199041015840 The transforma-

tion matrix from s frame to 1199041015840 frame can be acquired by 120575120579120575120574 and then the output attitude matrix 119862119899

119887

can be updated by

119862

119899

119887

= 119862

119899

1199041015840119862

119904

1015840

119904

119862

119904

119887

(10)

Figure 12 shows the relationship between s frame and 1199041015840

frameThe transformationmatrix1198621199041015840

119904

can be obtained by twoEuler angle rotations according to

119862

119904

1015840

119904

= 119877

119910(120575120574) 119877

119909(120575120579)

(11)

where 119877

119910(120575120574) = [

cos 120575120574 0 minus sin 1205751205740 1 0

sin 120575120574 0 cos 120575120574] and 119877

119909(120575120579) =

[

1 0 00 cos 120575120579 sin 1205751205790 minus sin 120575120579 cos 120575120579

]Since deviation angles 120575120579 120575120574 between 119904 frame and 119904

1015840

frame are small the matrix 1198621199041015840

119904

can be simplified as follows

119862

119904

1015840

119904

asymp

[

[

[

1 0 minus120575120574

0 1 120575120579

120575120574 minus120575120579 1

]

]

]

(12)

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

OriginalFitted

Angle (∘)

Enco

der a

ngle

erro

r (998400998400

)

Figure 13 Relationship between encoder angle error and rotationangle

4 RINS Attitude Error Compensation andExperimental Verification

According to the previous analysis both encoder angle errorsand axis irregular rotation can be corrected and then RINSattitude error fluctuation can be mostly compensated thusimproving short-term attitude output accuracy

41 AzimuthOutput Correction Experimental Results in RINSIt can be seen in Figure 8 that 120575120593 is axial symmetry aboutforward and reverse rotation thus it is closely related toencoder angle 120593 and the relationship between 120575120593 and 120593 isshown in Figure 13

Figure 13 indicates that the fluctuation of 120575120593 presents fun-damental frequencywith respect to the rotation angle Conse-quently the error variation can be compensated through datafitting by the mathematical model given in

120575120593 = 1198960 + 1198961 sin120593+ 1198962 cos120593 (13)

Use 119883 to denote state variable where 119883 = [1198960 1198961 1198962]119879

and then the coefficients 1198960 1198961 1198962 can be acquired throughleast square fitting algorithm by the following measurementsequation

119885 = 119867119883 (14)

where 119885 = [1205751205931 1205751205932 sdot sdot sdot 120575120593

119898]

119879 119867 = [

1 sin1205931 cos12059311 sin1205932 cos1205932sdotsdotsdot sdotsdotsdot sdotsdotsdot

1 sin120593119898

cos120593119898

]

119898times3

and119898 denotes the total number of useful measurementsSince the coefficients 1198960 1198961 1198962 fitting results are acquired

(in this RINS prototype the coefficients fitting results are1198960 = 0032 1198961 = minus6647 and 1198962 = 1249) the encoder angleerror in RINS could be corrected in real time by correctingcoefficients 1198960 1198961 1198962 according to

8 Mathematical Problems in Engineering

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

Angle (∘)

Resid

uals

(998400998400)

Figure 14 Fitting residuals of encoder angle errors

0 500 1000 1500 2000 2500 3000 3500 4000minus120

minus80

minus40

0

40

80

120

160

Time (s)

OriginalCompensated

Azi

mut

h er

ror (

998400998400)

Figure 15 Comparison of original and compensated azimutherrors

120593

119862= 120593minus (1198960 + 1198961 sin120593+ 1198962 cos120593) (15)

where 120593119862denotes the encoder angle to be compensated

Figure 14 shows the fitting residuals of encoder angleerrors and it is less than 2010158401015840 Then the transformation matrix119862

119904

119887

can be calculated by 120593119862instead of 120593 The azimuth output

correction results and comparisons are shown in Figure 15It can be seen from Figure 15 that after azimuth out-

put correction the peak-to-peak azimuth errors decreasedfrom 2sim3 arc minutes to less than 05 arc minutes whichis improved nearly 5 times and it means the short-termazimuth output accuracy achieves significantly improvement

42 Pitch and Roll Compensation Results in RINS The actualexperimental data also show that being similar to encoderangle errors the derivation angles 120575120579 120575120574 acquired by resid-uals of horizontal accelerometers are axial symmetry withrespect to encoder angle 120593 which are shown in Figure 16 and

Table 2 Coefficients estimation results (10158401015840)

1198861 1198871 1198862 1198872

120575120579 minus4209 minus6664 7793 6057120575120574 minus6414 4108 minus0127 minus9708

they can be used to correct and compensate pitch and rolloutput

From Figure 16 it can be seen that the fluctuation of 120575120579120575120574mainly presents second harmonic frequency with respectto rotation period Consequently it can be modeled by (16)and the coefficients estimation results are shown in Table 2Consider

120575120579 = 119886

1205791 sin120593+ 1198871205791 cos120593+ 1198861205792 sin 2120593+ 1198871205792 cos 2120593

120575120574 = 119886

1205741 sin120593+ 1198871205741 cos120593+ 1198861205742 sin 2120593+ 1198871205742 cos 2120593(16)

Figure 17 shows the fitting residuals of 120575120579 120575120574 and theyare less than 510158401015840 Then the transformation matrix 119862119904

119887

can becalculated according to (10) and (11) Thus the pitch and rolloutput compensation results can be acquired as shown inFigure 18

Conclusions can be drawn from Figure 18 that after pitchand roll output compensation the peak-to-peak pitch and rollerrors decreased from 20sim30 arc seconds to less than 5 arcseconds which is improved nearly 5 times It is proved thatthe proposed output compensation algorithm for pitch androll is valid and the pitch and roll short-term output accuracyis improved obviously

However it is worth mentioning that the compensationalgorithm here is not only suitable for single-axis rotationRINS For dual-axis and tri-axis RINS it is inevitable tohave encoder installation eccentricity and other mentionedproblems meanwhile the irregular rotation is hard to avoidas well consequently the analysis and attitude compensationand correction algorithm presented in this paper are alsosuitable for other types of RINS

5 Conclusion

This paper researched the attitude output accuracy improve-ment in rotation RINS Comparative experiment resultsindicate that velocity and position accuracy gains greatimprovement in rotation mode while attitude accuracy iseven worse than that in strapdown mode The reasons thatcause attitude output accuracy loss are analyzed and thena new attitude output compensation algorithm for RINS ispresented The experimental results proved validity of theproposed attitude correction and compensation algorithmwith short-term pitch and roll output accuracy improvedfrom 20sim30 arc seconds to less than 5 arc seconds andazimuth output accuracy improved from 2sim3 arc minutes toless than 05 arc minutes

According to dead reckoning principle high accuracy invelocity and position should be matched with high attitudeaccuracy The proposed compensation algorithm in thispaper solved the problem of attitude accuracy loss of RINSin practical applications which is significant for many task

Mathematical Problems in Engineering 9

0 60 120 180 240 300 360minus20

0

20

40

OriginalFitted

OriginalFitted

0 60 120 180 240 300 360minus20

0

20

40

120575120574

(998400998400)

120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 16 Relationships between 120575120579 120575120574 and rotation angle

0 60 120 180 240 300 360minus20

0

20

40

0 60 120 180 240 300 360minus20

0

20

40

Resid

uals

of120575120574

(998400998400)

Resid

uals

of120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 17 Fitting residuals of 120575120579 120575120574

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

OriginalCompensated

OriginalCompensated

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

Pitc

h er

ror (

998400998400)

Roll

erro

r (998400998400

)

Figure 18 Comparison of original and compensated pitch and roll errors

systems where attitude accuracy is urgently required Inaddition the proposed compensation algorithm is a generalmethod it is not only suitable for single-axis RINS but it canalso be used to improve short-term attitude output accuracyin dual-axis and tri-axis RINS

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported and funded by the National Nat-ural Science Foundation of China (L142200032) and Long-Term Development Strategic Research of China EngineeringScience and Technology (2014-zcq-01) The authors appreci-ate the support and fund

References

[1] E S Geller ldquoInertial system platform rotationrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol AES-4 no 4 pp557ndash568 1968

[2] E Levinson and R Majure ldquoAccuracy enhancement techniquesapplied to the marine ring laser inertial navigator (MARLIN)rdquoNavigation vol 34 no 1 pp 64ndash86 1987

[3] E Levinson J ter Horst and M Willcocks ldquoThe next genera-tion marine inertial navigator is here nowrdquo in Proceedings of thePosition Location and Navigation Symposium pp 121ndash127 IEEEApril 1994

[4] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation Journal ofthe Institute of Navigation vol 28 no 4 pp 311ndash341 1981

[5] Y Yang and L-J Miao ldquoFiber-optic strapdown inertial systemwith sensing cluster continuous rotationrdquo IEEE Transactions onAerospace and Electronic Systems vol 40 no 4 pp 1173ndash11782004

10 Mathematical Problems in Engineering

[6] S Ishibashi S Tsukioka H Yoshida et al ldquoAccuracy improve-ment of an inertial navigation system brought about by therotational motionrdquo in Proceedings of the OCEANSmdashEurope pp1ndash5 IEEE Aberdeen Scotland June 2007

[7] S Ishibashi S Tsukioka T Sawa et al ldquoThe rotation controlsystem to improve the accuracy of an inertial navigation systeminstalled in an autonomous underwater vehiclerdquo in Proceedingsof the Symposium on Underwater Technology and Workshop onScientific Use of Submarine Cables and Related Technologies pp495ndash498 IEEE Tokyo Japan April 2007

[8] A Li G-B Chang F-J Qin and H-W Li ldquoImproved precisionof strapdown inertial navigation system brought by dual-axiscontinuous rotation of inertial measurement unitrdquo in Proceed-ings of the 2nd International Asia Conference on Informatics inControl Automation and Robotics (CAR rsquo10) vol 1 pp 284ndash287March 2010

[9] G Chang J Xu A Li and K Cheng ldquoError analysis andsimulation of the dual-axis rotation-dwell autocompensatingstrapdown inertial navigation systemrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo10) pp 124ndash127 March 2010

[10] Z Deng M Sun and B Wang ldquoError modulation schemeanalysis of dual-axis rotating strap-down inertial navigationsystem based on FOGrdquo in Proceedings of the 33rd ChineseControl Conference (CCC rsquo14) pp 692ndash696Nanjing China July2014

[11] C Jianhua LMingyue CDaidai C Li and S Junyu ldquoResearchof strapdown inertial navigation system monitor techniquebased on dual-axis consequential rotationrdquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 203ndash208 June 2011

[12] R B Morrow Jr and D W Heckman ldquoHigh precision IFOGinsertion into the strategic submarine navigation systemrdquo inProceedings of the IEEE Position Location and Navigation Sym-posium pp 332ndash338 IEEE Palm Springs Calif USA April1998

[13] D W Heckman and L M Baretela ldquoImproved affordabilityof high precision submarine inertial navigation by insertion ofrapidly developing fiber optic gyro technologyrdquo in Proceedingsof the IEEE Position Location and and Navigation Symposiumpp 404ndash410 March 2000

[14] S Qin Z Huang and X Wang ldquoOptical angular encoderinstallation error measurement and calibration by ring lasergyroscoperdquo IEEETransactions on Instrumentation andMeasure-ment vol 59 no 3 pp 506ndash511 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Encoder

Torque motor

Az

Gz

Gy AyGx

Ax

Oys

xs

Figure 1 System structure

scheme [11] inertial sensor errors along three directions canbe restricted simultaneously Besides the USA has startedthe IFOG strategic nuclear submarine plan and expected tomanufacture the first tri-axis RINS system in 2010 [12] Thetri-axis RINS system is designed with three-axis continuousrotation scheme [13] and can compensate the scale factorerror of sensors installation axis instability gyroscope driftsand so forth These papers demonstrate that RINS has beenwidely researched and has huge prospect for improving INSnavigation accuracy However most research on RINS aimsat velocity and position accuracy improvement which isalso verified by RINS experimental results in this paperBut it should be noticed that RINS experimental resultsalso show that short-term attitude output accuracy of RINSgets even worse compared with that of nonrotation INSIn some airborne shipborne or land vehicles applicationshigh accuracy attitude information output is more crucial forsome task systems and loss of attitude accuracy will lead toserious consequences Since INS is based on dead reckoningprinciple high accuracy in position and velocity should bematched with attitude accuracy This paper researched thereasons that caused the loss of RINS attitude output accuracyand then presented the corresponding attitude accuracyimprovement algorithm Section 2 described the RINS sys-tem configuration and analyzed errors restraint principle andpreliminary experiment results Section 3 presented RINSattitude update algorithm and researched the possible rea-sons of attitude output accuracy loss Section 4 proposedthe designed RINS attitude compensation algorithm andpresented the experimental verification results Conclusionswere drawn in Section 5

2 RINS Error Restraint Analysis andExperiment Verification

21 RINS Error Restraint Analysis The system structure ofthe experimental RINS is shown in Figure 1The IMUmainlyincludes three Ring LaserGyroscopes (RLG) and three quartzaccelerometersThe torque motor drives IMU to rotate alongthe vertical body axis bidirectionally and an angular encoderis fixed to provide rotation angle relative to the body frame

Table 1 Main componentsrsquo specification of RINS

Gyro drifts 0008∘hAccelerometers biases 50 120583gEncoder resolution 1081015840pulseRotation angular rate 0686∘s

O

ys

120593

120593

xbxs

yb

zb(zs)

Figure 2 Frame definition

Table 1 shows the main componentsrsquo specification of theexperimental RINS

The proposed encoder provides digital output of 20000pulses per 360∘ rotation and the resolution is 1081015840pulseThe experimental RINS takes the scheme of bidirectionallycontinuous rotation and the rotation period is 1050 s Con-sidering IMU rotation the experimental RINS involves a newframe named inertial sensing frame (119904 frame O-xsyszs) Asis shown in Figure 2 119904 frame refers to the rotation framewhich varies with the change of inertial sensorsrsquo pointingdirection in real time The body frame (119887 frame O-xbybzbright-forward-upward) is defined the same as 119904 frame whenrotation angle 120593 is zero

The errors restraint principle of RINS can be explained bythe following analysis At time 119905 the angular rate and specificforce output of the gyroscopes and accelerometers are

120596

119904

119894119904

= 119862

119904

119887

120596

119887

119894119887

+ [120576

119909120576

119910120576

119911]

119879

+120596

119904

119887119904

119891

119904

119894119904

= 119862

119904

119887

119891

119887

119894119887

+ [nabla

119909nabla

119910nabla

119911]

119879

(1)

where 120596119887119894119887

and 119891

119887

119894119887

denote ideal gyroscopes and accelerom-eters outputs along 119887 frame 119862119887

119904

denotes the coordinate-transformationmatrix between 119887 frame and 119904 frame 120576

119909 120576119910 120576119911

are gyroscope drifts and nabla119909 nabla119910 nabla119911are accelerometer biases

120596

119904

119887119904

= [0 0 120596]

119879 represents rotation rate projected on 119904 frameAfterwards the angular rate and the specific force in body

frame can be obtained by the following equations

119887

119894119887

= 119862

119887

119904

120596

119904

119894119904

+120596

119887

119904119887

119891

119887

119894119887

= 119862

119887

119904

119891

119904

119894119904

(2)

Mathematical Problems in Engineering 3

where

119887

119894119887

and

119891

119887

119894119887

represent gyroscope and accelerometeractual outputs and 120596

119887

119904119887

= minus119862

119887

119904

120596

119904

119887119904

= [0 0 minus120596]

119879 Then (3)can be obtained by substituting (1) into (2) as follows

[

[

[

[

119887

119894119887119909

119887

119894119887119910

119887

119894119887119911

]

]

]

]

= 120596

119887

119894119887

+

[

[

[

120576

119909cos120593 minus 120576

119910sin120593

120576

119909sin120593 + 120576

119910cos120593

120576

119911

]

]

]

[

[

[

[

119891

119887

119894119887119909

119891

119887

119894119887119910

119891

119887

119894119887119911

]

]

]

]

= 119891

119887

119894119887

+

[

[

[

nabla

119909cos120593 minus nabla

119910sin120593

nabla

119909sin120593 + nabla

119910cos120593

nabla

119911

]

]

]

(3)

where 120593 denotes the rotation angle between 119887 frame and 119904

frame provided by the encoder If the rotation rate of IMU isideal the rotation angle 120593 equals 120596119905 during forward rotationand minus120596119905 during reverse rotation

In traditional strapdown INS constant inertial sensorerrors (gyro drifts and accelerometer biases) in 119887 frame arethe main contributor to INS positioning error while in RINSinertial sensor errors turn into periodical components Thenafter dead reckoning calculation attitude errors horizontalvelocity errors and horizontal position errors caused bygyroscope drifts and accelerometer biases can be modulatedso navigation errors in RINS are no longer accumulated withtime and navigation accuracy could achieve great improve-ment

22 RINS Experimental Verification Results In order tocompare the error restraint effect brought about by IMUrotation the experimental RINS can be transformed intotraditional strapdown INS by locking the torque motorrotation axis while navigation algorithm remains unchangedUnder the sameworking condition comparative experimentsare conducted in both rotation and strapdown mode whichlast 4200 s (4 bidirectional rotation periods) The proposedRINS prototype and associated experimental equipment andtheir connections are shown in Figure 3The RINS prototypeis mounted on a dual-axis turntable the experimental dataare collected by a laptop

Experimental results are given as follows Figure 4 showshorizontal position errors Figure 5 shows horizontal velocityerrors and Figure 6 shows attitude errors The dashed curverepresents strapdown results and the solid curve representsrotation results

In Figures 4 and 5 E is short for east and N is shortfor north The maximum of horizontal positioning errorin the strapdown mode is 1500m during experiment whilemaximum positioning error is only 160m in rotation modewhich means that positioning accuracy is improved nearly10 times in RINS The maximum of horizontal velocity errorin the strapdown mode is 15ms while maximum velocityerror in rotation mode is less than 03ms It means velocityaccuracy is improved about 5 timesThus it can be seen thatby driving IMU rotation bidirectionally horizontal positionand velocity accuracy are improved obviously in RINS

The attitude outputs during comparative experiment aregiven in Figure 6 It can be seen that in strapdown mode

Turntable

Power supply

RINS

Laptop

Figure 3 RINS and associated experimental equipment

attitude error changes slowly during experiment and it showsgood stability in short term (several minutes) The relativelylong-term periodical varying components in pitch and rollerrors are Schuler oscillations (844min period) Howeverin rotation mode attitude errors become more complicatedPitch and roll errors are stable on the general trend duringthe whole 70-minute experiment The mean value of attitudeerrors in every rotation period is nearly the same It is hardto see Schuler oscillations in the error curve because mostof horizontal inertial sensor errors have been attenuatedby IMU rotation thus Schuler oscillations amplitude issmall However the short-term periodical fluctuation arisesin attitude outputs with oscillations amplitude of 20sim30 arcseconds in pitch and roll errors and 2sim3 arc minutes inazimuth errors Compared to strapdown mode position andvelocity accuracy inRINS increased greatly but attitude errorsfail to reach the same accuracy in short term (even getworse) let alone improvement by IMU rotation But for someairborne shipborne or land vehicle task systems not onlyis high accuracy positioning and velocity reference neededbut also high accuracy attitude reference is important Errorfluctuation of 20 arc seconds in pitch and roll output anderror fluctuation of 2 arc minutes in azimuth output willhave seriously negative effect on their performance Howeveraccording to theoretical analysismentioned in Section 2 highaccuracy in position and velocity should be matched withhigh attitude accuracy since INS is based on dead reckoningprinciple Therefore it is very necessary to analyze causes ofthe attitude output fluctuation errors and to find solutions toimprove short-term attitude accuracy in RINS

3 Analysis of Attitude OutputAccuracy in RINS

31 AttitudeUpdate Algorithm in RINS Thebodyrsquos attitude inINS can be described by transformation matrix from 119887 frameto 119899 frame (navigation frame) through attitude matrix 119862119899

119887

In

4 Mathematical Problems in Engineering

minus2000

minus1000

0

1000

Time (s)

PE er

ror (

m)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 40000 500 1000 1500 2000 2500 3000 3500 4000minus500

0

500

1000

Time (s)

PN er

ror (

m)

RotationStrapdown

Figure 4 PE and PN errors comparison

0 500 1000 1500 2000 2500 3000 3500 4000minus2

minus1

0

1

Time (s)

VE

erro

r (m

s)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 4000minus05

0

05

1

Time (s)

VN

erro

r (m

s)

RotationStrapdown

Figure 5 VE and VN errors comparison

0 500 1000 1500 2000 2500 3000 3500 4000minus50

0

50

Time (s)

RotationStrapdown

Pitc

h er

ror (

998400998400)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 4000minus50

0

50

Time (s)

Roll

erro

r (998400998400

)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 4000minus200

0

200

Time (s)

Azi

mut

h er

ror (

998400998400)

Figure 6 Strapdown and rotation INS attitude errors comparison

strapdown INS configuration 119862119899119887

is calculated directly fromrotation vector by quaternion attitude updating algorithmHowever in RINS configuration attitude matrix 119862119899

119904

shouldbe updated beforehand (the update process ofmatrix119862119899

119904

is thesame as that of119862119899

119887

in strapdown INS) then119862119899119887

can be acquiredaccording to the following equation

119862

119899

119887

= 119862

119899

119904

119862

119904

119887

(4)

The matrix 119862119904119887

is the transformation matrix from 119887 frameto 119904 frame and is given as follows

119862

119904

119887

=

[

[

[

cos120593 sin120593 0minus sin120593 cos120593 0

0 0 1

]

]

]

(5)

where 120593 is the rotation angle provided by encoder

Mathematical Problems in Engineering 5

If all the updating processes above are ideal attitudeoutput in RINS should present high accuracy and shouldnot contain short-term error fluctuation As a matter of factin RINS updating matrix 119862

119899

119904

is calculated from rotationvector by quaternion attitude updating algorithm which isthe same as 119862119899

119887

in strapdown INS thus it is accurate enoughConsequently the most possible cause of RINS short-termattitude output accuracy loss is transformation process fromattitude matrix 119862119899

119904

to 119862119899119887

that is to say matrix 119862119904119887

In actualRINS the encoder output errors installation eccentricity ofencoder and noncoaxial rotation of IMU can cause rotationangle 120593 containing periodical output angle errors [14] whichlead to azimuth output fluctuation inRINS Besides if the axisO-zs does not coincidewith the rotation axisO-zb all the timethe rotation angular rate will have projection along horizontalaxes O-xs and O-ys Thus undesirable influences will beattached to the transformation matrix 119862119904

119887

which will resultin error fluctuation on pitch and roll output The detailedanalysis will be given as follows

32 Analysis of Azimuth Output Error in RINS Equation (5)indicates that the rotation angle 120593 will have effect on theaccuracy of matrix 119862119904

119887

Therefore it is necessary to analyzethe rotation angle provided by the encoder first In orderto obtain the encoder angle errors the angle integrated byoutput angular rate of gyroscope 119885 is used However outputof gyroscope 119885 contains not only rotation angular rate 120596 butalso gyroscope drift 120576

119911and the upward component of earth

rotation angular rate 120596119894119890sin 119871 The sign of IMU rotation rate

120596 is opposite during forward and reverse rotation Then thegyroscope outputs are given as

120596

119911119891= 120596+120596

119894119890sin 119871+ 120576

119911

120596

119911119903= minus120596+120596

119894119890sin 119871+ 120576

119911

(6)

where subscript 119891 denotes forward rotation and 119903 denotesreverse rotation If we define 1205961015840 as 120596

119894119890sin 119871 + 120576

119911= (120596

119911119891+

120596

119911119903)2 then the angle integrated by output angular rate

of gyroscope 119885 during IMU rotation can be calculatedaccording to (7) which will be a reference for encoder angleoutput Consider

120593

119911119891= int (120596

119911119891minus120596

1015840

) 119889119905

120593

119911119903= int (120596

119911119903minus120596

1015840

) 119889119905

(7)

Figure 7 shows actual experimental data of encoder angleoutputs and Figure 8 is acquired when subtracting encoderangle from the integrated angle of gyroscope 119885 It can beseen from the two figures that encoder angle error (defineas 120575120593) presents obvious repeatability during 4 bidirectionalrotation periods and the fluctuations are axial symmetryabout forward and reverse rotation in a separate bidirectionalperiod 120575120593 presents periodical fluctuation with the sameperiod as IMU rotation and the fluctuation amplitude is 1sim2 arc minutes It presents the same property as azimuth errorin Figure 6 and can be compensated afterwards short-termazimuth output accuracy would be greatly improved

0 500 1000 1500 2000 2500 3000 3500 40000

50

100

150

200

250

300

350

400

Time (s)

Enco

der o

utpu

t (∘ )

Figure 7 Encoder angular output

0 500 1000 1500 2000 2500 3000 3500 4000minus150

minus100

minus50

0

50

100

150

Time (s)

Enco

der a

ngle

erro

r (998400998400

)

Figure 8 Encoder angular error

33 Analysis of Pitch and Roll Output Error in RINS Theexpression of matrix 119862

119904

119887

in (5) is obtained based on idealrotation namely the axes of O-xs and O-ys stay in a fixedplane during rotation However in fact irregular rotation ishard to avoid and the rotation planewould change in differentpositions The difference between actual rotation and idealrotation should be analyzed in detail

Figure 9 (left) describes IMU rotation under ideal condi-tion and in this case axes of O-xs and O-ys in 119904 frame remainunchanged in fixed plane and axis O-zs coincides with O-zbin body frame all the time However in actual condition dueto some reasons such as machining and assembling errors inrotation axis and defects in rotation bearings the IMU rotatesirregularly Figure 9 (right) shows this irregular rotation andthe rotation plane in 119904 frame no longer stays unchanged thepointing of O-zs axis will change in space and axes of O-xsand O-ys will fluctuate around the ideal fixed plane

If the rotation plane remains unchanged the projectionof acceleration of gravity on horizontal accelerometers will

6 Mathematical Problems in Engineering

Actual rotation axis

Rotation axis

ys

Ay

Az

xs

xs

AxAx

zszb

ysAz

Ay 120575120579

120575120574

zb(zs)

Figure 9 Comparison of ideal rotation and actual irregular rotation

0 500 1000 1500 2000 2500 3000 3500 4000minus004

minus002

0

002

004

Time (s)0 500 1000 1500 2000 2500 3000 3500 4000

minus004

minus002

0

002

004

Time (s)

Acce

lero

met

er x

(ms2)

Acce

lero

met

er y

(ms2)

Figure 10 Initial outputs of horizontal accelerometers

0 500 1000 1500 2000 2500 3000 3500 4000minus200

minus100

0

100

200

Time (s)0 500 1000 1500 2000 2500 3000 3500 4000

minus200

minus100

0

100

200

Time (s)

Resid

ual o

f acc

elero

met

er120583

g)

Resid

ual o

f acc

eler

omet

er120583

g)

x (

y (

Figure 11 Residual components of horizontal accelerometers

be standard sine or cosine form with the same period asIMU rotation and the amplitude is determined by initialpitch and roll of the rotation plane However due to thefluctuation of rotation plane in actual situation the projectionof acceleration of gravity on horizontal accelerometers willbring about irregular variation and the deviation betweenthe rotation plane and the fixed plane can be representedby irregular variation in horizontal accelerometers Figure 10shows the original output of two horizontal accelerometers inactual experiments In order to obtain the irregular rotationcomponents firstly the projection of acceleration of gravitycaused by pitch and roll should be deducted from thehorizontal accelerometers according to

119886

1015840

119909

= 119886

119909+1198921205740 cos120593minus1198921205790 sin120593

119886

1015840

119910

= 119886

119910minus1198921205790 cos120593minus1198921205740 sin120593

(8)

where 1205790 1205740 are initial pitch and roll acquired by thealignment process in RINS

Figure 11 shows the residual components of horizon-tal accelerometers during the experiment Both horizon-tal accelerometers residual components mainly representsecond harmonic frequency related to the rotation periodand show good repeatability during 4 bidirectional rotationperiods In a single bidirectional period the fluctuationson the horizontal accelerometers are also axial symmetryThe fluctuation amplitude is about 150 120583g which means themaximum deviation between the rotation plane and the fixedplane is about 30 arc seconds and it coincides with theamplitude of pitch and roll output fluctuation in Figure 6

The residuals of horizontal accelerometers can be trans-formed into deviation angles between the rotation plane and

Mathematical Problems in Engineering 7

120575120574

120575120574

120575120579

120575120579O

zs

xs

ys

y998400s

x998400s

z998400s

Figure 12 Transformation relation between s frame and 1199041015840 frame

the fixed plane which are defined as 120575120579 120575120574 as shown inFigure 9 (right) and are given as follows

120575120579 =

119886

1015840

119910

119892

120575120574 = minus

119886

1015840

119909

119892

(9)

To compensate influence caused by 120575120579 120575120574 anothercoordinate frame should be defined in the proposed RINS todistinguish actual irregular rotation from ideal rotation Theideal rotation frame is named as 1199041015840 frame whose O-1199111015840

119904

axis iscoinciding with O-zb axis all the time The previous 119904 frameis the actual rotation frame representing the instantaneousrotation frame and it fluctuates around the ideal fixed planeTaking into consideration irregular rotation attitude matrixobtained by (6) should be described as 119862119899

1199041015840 The transforma-

tion matrix from s frame to 1199041015840 frame can be acquired by 120575120579120575120574 and then the output attitude matrix 119862119899

119887

can be updated by

119862

119899

119887

= 119862

119899

1199041015840119862

119904

1015840

119904

119862

119904

119887

(10)

Figure 12 shows the relationship between s frame and 1199041015840

frameThe transformationmatrix1198621199041015840

119904

can be obtained by twoEuler angle rotations according to

119862

119904

1015840

119904

= 119877

119910(120575120574) 119877

119909(120575120579)

(11)

where 119877

119910(120575120574) = [

cos 120575120574 0 minus sin 1205751205740 1 0

sin 120575120574 0 cos 120575120574] and 119877

119909(120575120579) =

[

1 0 00 cos 120575120579 sin 1205751205790 minus sin 120575120579 cos 120575120579

]Since deviation angles 120575120579 120575120574 between 119904 frame and 119904

1015840

frame are small the matrix 1198621199041015840

119904

can be simplified as follows

119862

119904

1015840

119904

asymp

[

[

[

1 0 minus120575120574

0 1 120575120579

120575120574 minus120575120579 1

]

]

]

(12)

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

OriginalFitted

Angle (∘)

Enco

der a

ngle

erro

r (998400998400

)

Figure 13 Relationship between encoder angle error and rotationangle

4 RINS Attitude Error Compensation andExperimental Verification

According to the previous analysis both encoder angle errorsand axis irregular rotation can be corrected and then RINSattitude error fluctuation can be mostly compensated thusimproving short-term attitude output accuracy

41 AzimuthOutput Correction Experimental Results in RINSIt can be seen in Figure 8 that 120575120593 is axial symmetry aboutforward and reverse rotation thus it is closely related toencoder angle 120593 and the relationship between 120575120593 and 120593 isshown in Figure 13

Figure 13 indicates that the fluctuation of 120575120593 presents fun-damental frequencywith respect to the rotation angle Conse-quently the error variation can be compensated through datafitting by the mathematical model given in

120575120593 = 1198960 + 1198961 sin120593+ 1198962 cos120593 (13)

Use 119883 to denote state variable where 119883 = [1198960 1198961 1198962]119879

and then the coefficients 1198960 1198961 1198962 can be acquired throughleast square fitting algorithm by the following measurementsequation

119885 = 119867119883 (14)

where 119885 = [1205751205931 1205751205932 sdot sdot sdot 120575120593

119898]

119879 119867 = [

1 sin1205931 cos12059311 sin1205932 cos1205932sdotsdotsdot sdotsdotsdot sdotsdotsdot

1 sin120593119898

cos120593119898

]

119898times3

and119898 denotes the total number of useful measurementsSince the coefficients 1198960 1198961 1198962 fitting results are acquired

(in this RINS prototype the coefficients fitting results are1198960 = 0032 1198961 = minus6647 and 1198962 = 1249) the encoder angleerror in RINS could be corrected in real time by correctingcoefficients 1198960 1198961 1198962 according to

8 Mathematical Problems in Engineering

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

Angle (∘)

Resid

uals

(998400998400)

Figure 14 Fitting residuals of encoder angle errors

0 500 1000 1500 2000 2500 3000 3500 4000minus120

minus80

minus40

0

40

80

120

160

Time (s)

OriginalCompensated

Azi

mut

h er

ror (

998400998400)

Figure 15 Comparison of original and compensated azimutherrors

120593

119862= 120593minus (1198960 + 1198961 sin120593+ 1198962 cos120593) (15)

where 120593119862denotes the encoder angle to be compensated

Figure 14 shows the fitting residuals of encoder angleerrors and it is less than 2010158401015840 Then the transformation matrix119862

119904

119887

can be calculated by 120593119862instead of 120593 The azimuth output

correction results and comparisons are shown in Figure 15It can be seen from Figure 15 that after azimuth out-

put correction the peak-to-peak azimuth errors decreasedfrom 2sim3 arc minutes to less than 05 arc minutes whichis improved nearly 5 times and it means the short-termazimuth output accuracy achieves significantly improvement

42 Pitch and Roll Compensation Results in RINS The actualexperimental data also show that being similar to encoderangle errors the derivation angles 120575120579 120575120574 acquired by resid-uals of horizontal accelerometers are axial symmetry withrespect to encoder angle 120593 which are shown in Figure 16 and

Table 2 Coefficients estimation results (10158401015840)

1198861 1198871 1198862 1198872

120575120579 minus4209 minus6664 7793 6057120575120574 minus6414 4108 minus0127 minus9708

they can be used to correct and compensate pitch and rolloutput

From Figure 16 it can be seen that the fluctuation of 120575120579120575120574mainly presents second harmonic frequency with respectto rotation period Consequently it can be modeled by (16)and the coefficients estimation results are shown in Table 2Consider

120575120579 = 119886

1205791 sin120593+ 1198871205791 cos120593+ 1198861205792 sin 2120593+ 1198871205792 cos 2120593

120575120574 = 119886

1205741 sin120593+ 1198871205741 cos120593+ 1198861205742 sin 2120593+ 1198871205742 cos 2120593(16)

Figure 17 shows the fitting residuals of 120575120579 120575120574 and theyare less than 510158401015840 Then the transformation matrix 119862119904

119887

can becalculated according to (10) and (11) Thus the pitch and rolloutput compensation results can be acquired as shown inFigure 18

Conclusions can be drawn from Figure 18 that after pitchand roll output compensation the peak-to-peak pitch and rollerrors decreased from 20sim30 arc seconds to less than 5 arcseconds which is improved nearly 5 times It is proved thatthe proposed output compensation algorithm for pitch androll is valid and the pitch and roll short-term output accuracyis improved obviously

However it is worth mentioning that the compensationalgorithm here is not only suitable for single-axis rotationRINS For dual-axis and tri-axis RINS it is inevitable tohave encoder installation eccentricity and other mentionedproblems meanwhile the irregular rotation is hard to avoidas well consequently the analysis and attitude compensationand correction algorithm presented in this paper are alsosuitable for other types of RINS

5 Conclusion

This paper researched the attitude output accuracy improve-ment in rotation RINS Comparative experiment resultsindicate that velocity and position accuracy gains greatimprovement in rotation mode while attitude accuracy iseven worse than that in strapdown mode The reasons thatcause attitude output accuracy loss are analyzed and thena new attitude output compensation algorithm for RINS ispresented The experimental results proved validity of theproposed attitude correction and compensation algorithmwith short-term pitch and roll output accuracy improvedfrom 20sim30 arc seconds to less than 5 arc seconds andazimuth output accuracy improved from 2sim3 arc minutes toless than 05 arc minutes

According to dead reckoning principle high accuracy invelocity and position should be matched with high attitudeaccuracy The proposed compensation algorithm in thispaper solved the problem of attitude accuracy loss of RINSin practical applications which is significant for many task

Mathematical Problems in Engineering 9

0 60 120 180 240 300 360minus20

0

20

40

OriginalFitted

OriginalFitted

0 60 120 180 240 300 360minus20

0

20

40

120575120574

(998400998400)

120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 16 Relationships between 120575120579 120575120574 and rotation angle

0 60 120 180 240 300 360minus20

0

20

40

0 60 120 180 240 300 360minus20

0

20

40

Resid

uals

of120575120574

(998400998400)

Resid

uals

of120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 17 Fitting residuals of 120575120579 120575120574

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

OriginalCompensated

OriginalCompensated

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

Pitc

h er

ror (

998400998400)

Roll

erro

r (998400998400

)

Figure 18 Comparison of original and compensated pitch and roll errors

systems where attitude accuracy is urgently required Inaddition the proposed compensation algorithm is a generalmethod it is not only suitable for single-axis RINS but it canalso be used to improve short-term attitude output accuracyin dual-axis and tri-axis RINS

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported and funded by the National Nat-ural Science Foundation of China (L142200032) and Long-Term Development Strategic Research of China EngineeringScience and Technology (2014-zcq-01) The authors appreci-ate the support and fund

References

[1] E S Geller ldquoInertial system platform rotationrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol AES-4 no 4 pp557ndash568 1968

[2] E Levinson and R Majure ldquoAccuracy enhancement techniquesapplied to the marine ring laser inertial navigator (MARLIN)rdquoNavigation vol 34 no 1 pp 64ndash86 1987

[3] E Levinson J ter Horst and M Willcocks ldquoThe next genera-tion marine inertial navigator is here nowrdquo in Proceedings of thePosition Location and Navigation Symposium pp 121ndash127 IEEEApril 1994

[4] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation Journal ofthe Institute of Navigation vol 28 no 4 pp 311ndash341 1981

[5] Y Yang and L-J Miao ldquoFiber-optic strapdown inertial systemwith sensing cluster continuous rotationrdquo IEEE Transactions onAerospace and Electronic Systems vol 40 no 4 pp 1173ndash11782004

10 Mathematical Problems in Engineering

[6] S Ishibashi S Tsukioka H Yoshida et al ldquoAccuracy improve-ment of an inertial navigation system brought about by therotational motionrdquo in Proceedings of the OCEANSmdashEurope pp1ndash5 IEEE Aberdeen Scotland June 2007

[7] S Ishibashi S Tsukioka T Sawa et al ldquoThe rotation controlsystem to improve the accuracy of an inertial navigation systeminstalled in an autonomous underwater vehiclerdquo in Proceedingsof the Symposium on Underwater Technology and Workshop onScientific Use of Submarine Cables and Related Technologies pp495ndash498 IEEE Tokyo Japan April 2007

[8] A Li G-B Chang F-J Qin and H-W Li ldquoImproved precisionof strapdown inertial navigation system brought by dual-axiscontinuous rotation of inertial measurement unitrdquo in Proceed-ings of the 2nd International Asia Conference on Informatics inControl Automation and Robotics (CAR rsquo10) vol 1 pp 284ndash287March 2010

[9] G Chang J Xu A Li and K Cheng ldquoError analysis andsimulation of the dual-axis rotation-dwell autocompensatingstrapdown inertial navigation systemrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo10) pp 124ndash127 March 2010

[10] Z Deng M Sun and B Wang ldquoError modulation schemeanalysis of dual-axis rotating strap-down inertial navigationsystem based on FOGrdquo in Proceedings of the 33rd ChineseControl Conference (CCC rsquo14) pp 692ndash696Nanjing China July2014

[11] C Jianhua LMingyue CDaidai C Li and S Junyu ldquoResearchof strapdown inertial navigation system monitor techniquebased on dual-axis consequential rotationrdquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 203ndash208 June 2011

[12] R B Morrow Jr and D W Heckman ldquoHigh precision IFOGinsertion into the strategic submarine navigation systemrdquo inProceedings of the IEEE Position Location and Navigation Sym-posium pp 332ndash338 IEEE Palm Springs Calif USA April1998

[13] D W Heckman and L M Baretela ldquoImproved affordabilityof high precision submarine inertial navigation by insertion ofrapidly developing fiber optic gyro technologyrdquo in Proceedingsof the IEEE Position Location and and Navigation Symposiumpp 404ndash410 March 2000

[14] S Qin Z Huang and X Wang ldquoOptical angular encoderinstallation error measurement and calibration by ring lasergyroscoperdquo IEEETransactions on Instrumentation andMeasure-ment vol 59 no 3 pp 506ndash511 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

where

119887

119894119887

and

119891

119887

119894119887

represent gyroscope and accelerometeractual outputs and 120596

119887

119904119887

= minus119862

119887

119904

120596

119904

119887119904

= [0 0 minus120596]

119879 Then (3)can be obtained by substituting (1) into (2) as follows

[

[

[

[

119887

119894119887119909

119887

119894119887119910

119887

119894119887119911

]

]

]

]

= 120596

119887

119894119887

+

[

[

[

120576

119909cos120593 minus 120576

119910sin120593

120576

119909sin120593 + 120576

119910cos120593

120576

119911

]

]

]

[

[

[

[

119891

119887

119894119887119909

119891

119887

119894119887119910

119891

119887

119894119887119911

]

]

]

]

= 119891

119887

119894119887

+

[

[

[

nabla

119909cos120593 minus nabla

119910sin120593

nabla

119909sin120593 + nabla

119910cos120593

nabla

119911

]

]

]

(3)

where 120593 denotes the rotation angle between 119887 frame and 119904

frame provided by the encoder If the rotation rate of IMU isideal the rotation angle 120593 equals 120596119905 during forward rotationand minus120596119905 during reverse rotation

In traditional strapdown INS constant inertial sensorerrors (gyro drifts and accelerometer biases) in 119887 frame arethe main contributor to INS positioning error while in RINSinertial sensor errors turn into periodical components Thenafter dead reckoning calculation attitude errors horizontalvelocity errors and horizontal position errors caused bygyroscope drifts and accelerometer biases can be modulatedso navigation errors in RINS are no longer accumulated withtime and navigation accuracy could achieve great improve-ment

22 RINS Experimental Verification Results In order tocompare the error restraint effect brought about by IMUrotation the experimental RINS can be transformed intotraditional strapdown INS by locking the torque motorrotation axis while navigation algorithm remains unchangedUnder the sameworking condition comparative experimentsare conducted in both rotation and strapdown mode whichlast 4200 s (4 bidirectional rotation periods) The proposedRINS prototype and associated experimental equipment andtheir connections are shown in Figure 3The RINS prototypeis mounted on a dual-axis turntable the experimental dataare collected by a laptop

Experimental results are given as follows Figure 4 showshorizontal position errors Figure 5 shows horizontal velocityerrors and Figure 6 shows attitude errors The dashed curverepresents strapdown results and the solid curve representsrotation results

In Figures 4 and 5 E is short for east and N is shortfor north The maximum of horizontal positioning errorin the strapdown mode is 1500m during experiment whilemaximum positioning error is only 160m in rotation modewhich means that positioning accuracy is improved nearly10 times in RINS The maximum of horizontal velocity errorin the strapdown mode is 15ms while maximum velocityerror in rotation mode is less than 03ms It means velocityaccuracy is improved about 5 timesThus it can be seen thatby driving IMU rotation bidirectionally horizontal positionand velocity accuracy are improved obviously in RINS

The attitude outputs during comparative experiment aregiven in Figure 6 It can be seen that in strapdown mode

Turntable

Power supply

RINS

Laptop

Figure 3 RINS and associated experimental equipment

attitude error changes slowly during experiment and it showsgood stability in short term (several minutes) The relativelylong-term periodical varying components in pitch and rollerrors are Schuler oscillations (844min period) Howeverin rotation mode attitude errors become more complicatedPitch and roll errors are stable on the general trend duringthe whole 70-minute experiment The mean value of attitudeerrors in every rotation period is nearly the same It is hardto see Schuler oscillations in the error curve because mostof horizontal inertial sensor errors have been attenuatedby IMU rotation thus Schuler oscillations amplitude issmall However the short-term periodical fluctuation arisesin attitude outputs with oscillations amplitude of 20sim30 arcseconds in pitch and roll errors and 2sim3 arc minutes inazimuth errors Compared to strapdown mode position andvelocity accuracy inRINS increased greatly but attitude errorsfail to reach the same accuracy in short term (even getworse) let alone improvement by IMU rotation But for someairborne shipborne or land vehicle task systems not onlyis high accuracy positioning and velocity reference neededbut also high accuracy attitude reference is important Errorfluctuation of 20 arc seconds in pitch and roll output anderror fluctuation of 2 arc minutes in azimuth output willhave seriously negative effect on their performance Howeveraccording to theoretical analysismentioned in Section 2 highaccuracy in position and velocity should be matched withhigh attitude accuracy since INS is based on dead reckoningprinciple Therefore it is very necessary to analyze causes ofthe attitude output fluctuation errors and to find solutions toimprove short-term attitude accuracy in RINS

3 Analysis of Attitude OutputAccuracy in RINS

31 AttitudeUpdate Algorithm in RINS Thebodyrsquos attitude inINS can be described by transformation matrix from 119887 frameto 119899 frame (navigation frame) through attitude matrix 119862119899

119887

In

4 Mathematical Problems in Engineering

minus2000

minus1000

0

1000

Time (s)

PE er

ror (

m)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 40000 500 1000 1500 2000 2500 3000 3500 4000minus500

0

500

1000

Time (s)

PN er

ror (

m)

RotationStrapdown

Figure 4 PE and PN errors comparison

0 500 1000 1500 2000 2500 3000 3500 4000minus2

minus1

0

1

Time (s)

VE

erro

r (m

s)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 4000minus05

0

05

1

Time (s)

VN

erro

r (m

s)

RotationStrapdown

Figure 5 VE and VN errors comparison

0 500 1000 1500 2000 2500 3000 3500 4000minus50

0

50

Time (s)

RotationStrapdown

Pitc

h er

ror (

998400998400)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 4000minus50

0

50

Time (s)

Roll

erro

r (998400998400

)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 4000minus200

0

200

Time (s)

Azi

mut

h er

ror (

998400998400)

Figure 6 Strapdown and rotation INS attitude errors comparison

strapdown INS configuration 119862119899119887

is calculated directly fromrotation vector by quaternion attitude updating algorithmHowever in RINS configuration attitude matrix 119862119899

119904

shouldbe updated beforehand (the update process ofmatrix119862119899

119904

is thesame as that of119862119899

119887

in strapdown INS) then119862119899119887

can be acquiredaccording to the following equation

119862

119899

119887

= 119862

119899

119904

119862

119904

119887

(4)

The matrix 119862119904119887

is the transformation matrix from 119887 frameto 119904 frame and is given as follows

119862

119904

119887

=

[

[

[

cos120593 sin120593 0minus sin120593 cos120593 0

0 0 1

]

]

]

(5)

where 120593 is the rotation angle provided by encoder

Mathematical Problems in Engineering 5

If all the updating processes above are ideal attitudeoutput in RINS should present high accuracy and shouldnot contain short-term error fluctuation As a matter of factin RINS updating matrix 119862

119899

119904

is calculated from rotationvector by quaternion attitude updating algorithm which isthe same as 119862119899

119887

in strapdown INS thus it is accurate enoughConsequently the most possible cause of RINS short-termattitude output accuracy loss is transformation process fromattitude matrix 119862119899

119904

to 119862119899119887

that is to say matrix 119862119904119887

In actualRINS the encoder output errors installation eccentricity ofencoder and noncoaxial rotation of IMU can cause rotationangle 120593 containing periodical output angle errors [14] whichlead to azimuth output fluctuation inRINS Besides if the axisO-zs does not coincidewith the rotation axisO-zb all the timethe rotation angular rate will have projection along horizontalaxes O-xs and O-ys Thus undesirable influences will beattached to the transformation matrix 119862119904

119887

which will resultin error fluctuation on pitch and roll output The detailedanalysis will be given as follows

32 Analysis of Azimuth Output Error in RINS Equation (5)indicates that the rotation angle 120593 will have effect on theaccuracy of matrix 119862119904

119887

Therefore it is necessary to analyzethe rotation angle provided by the encoder first In orderto obtain the encoder angle errors the angle integrated byoutput angular rate of gyroscope 119885 is used However outputof gyroscope 119885 contains not only rotation angular rate 120596 butalso gyroscope drift 120576

119911and the upward component of earth

rotation angular rate 120596119894119890sin 119871 The sign of IMU rotation rate

120596 is opposite during forward and reverse rotation Then thegyroscope outputs are given as

120596

119911119891= 120596+120596

119894119890sin 119871+ 120576

119911

120596

119911119903= minus120596+120596

119894119890sin 119871+ 120576

119911

(6)

where subscript 119891 denotes forward rotation and 119903 denotesreverse rotation If we define 1205961015840 as 120596

119894119890sin 119871 + 120576

119911= (120596

119911119891+

120596

119911119903)2 then the angle integrated by output angular rate

of gyroscope 119885 during IMU rotation can be calculatedaccording to (7) which will be a reference for encoder angleoutput Consider

120593

119911119891= int (120596

119911119891minus120596

1015840

) 119889119905

120593

119911119903= int (120596

119911119903minus120596

1015840

) 119889119905

(7)

Figure 7 shows actual experimental data of encoder angleoutputs and Figure 8 is acquired when subtracting encoderangle from the integrated angle of gyroscope 119885 It can beseen from the two figures that encoder angle error (defineas 120575120593) presents obvious repeatability during 4 bidirectionalrotation periods and the fluctuations are axial symmetryabout forward and reverse rotation in a separate bidirectionalperiod 120575120593 presents periodical fluctuation with the sameperiod as IMU rotation and the fluctuation amplitude is 1sim2 arc minutes It presents the same property as azimuth errorin Figure 6 and can be compensated afterwards short-termazimuth output accuracy would be greatly improved

0 500 1000 1500 2000 2500 3000 3500 40000

50

100

150

200

250

300

350

400

Time (s)

Enco

der o

utpu

t (∘ )

Figure 7 Encoder angular output

0 500 1000 1500 2000 2500 3000 3500 4000minus150

minus100

minus50

0

50

100

150

Time (s)

Enco

der a

ngle

erro

r (998400998400

)

Figure 8 Encoder angular error

33 Analysis of Pitch and Roll Output Error in RINS Theexpression of matrix 119862

119904

119887

in (5) is obtained based on idealrotation namely the axes of O-xs and O-ys stay in a fixedplane during rotation However in fact irregular rotation ishard to avoid and the rotation planewould change in differentpositions The difference between actual rotation and idealrotation should be analyzed in detail

Figure 9 (left) describes IMU rotation under ideal condi-tion and in this case axes of O-xs and O-ys in 119904 frame remainunchanged in fixed plane and axis O-zs coincides with O-zbin body frame all the time However in actual condition dueto some reasons such as machining and assembling errors inrotation axis and defects in rotation bearings the IMU rotatesirregularly Figure 9 (right) shows this irregular rotation andthe rotation plane in 119904 frame no longer stays unchanged thepointing of O-zs axis will change in space and axes of O-xsand O-ys will fluctuate around the ideal fixed plane

If the rotation plane remains unchanged the projectionof acceleration of gravity on horizontal accelerometers will

6 Mathematical Problems in Engineering

Actual rotation axis

Rotation axis

ys

Ay

Az

xs

xs

AxAx

zszb

ysAz

Ay 120575120579

120575120574

zb(zs)

Figure 9 Comparison of ideal rotation and actual irregular rotation

0 500 1000 1500 2000 2500 3000 3500 4000minus004

minus002

0

002

004

Time (s)0 500 1000 1500 2000 2500 3000 3500 4000

minus004

minus002

0

002

004

Time (s)

Acce

lero

met

er x

(ms2)

Acce

lero

met

er y

(ms2)

Figure 10 Initial outputs of horizontal accelerometers

0 500 1000 1500 2000 2500 3000 3500 4000minus200

minus100

0

100

200

Time (s)0 500 1000 1500 2000 2500 3000 3500 4000

minus200

minus100

0

100

200

Time (s)

Resid

ual o

f acc

elero

met

er120583

g)

Resid

ual o

f acc

eler

omet

er120583

g)

x (

y (

Figure 11 Residual components of horizontal accelerometers

be standard sine or cosine form with the same period asIMU rotation and the amplitude is determined by initialpitch and roll of the rotation plane However due to thefluctuation of rotation plane in actual situation the projectionof acceleration of gravity on horizontal accelerometers willbring about irregular variation and the deviation betweenthe rotation plane and the fixed plane can be representedby irregular variation in horizontal accelerometers Figure 10shows the original output of two horizontal accelerometers inactual experiments In order to obtain the irregular rotationcomponents firstly the projection of acceleration of gravitycaused by pitch and roll should be deducted from thehorizontal accelerometers according to

119886

1015840

119909

= 119886

119909+1198921205740 cos120593minus1198921205790 sin120593

119886

1015840

119910

= 119886

119910minus1198921205790 cos120593minus1198921205740 sin120593

(8)

where 1205790 1205740 are initial pitch and roll acquired by thealignment process in RINS

Figure 11 shows the residual components of horizon-tal accelerometers during the experiment Both horizon-tal accelerometers residual components mainly representsecond harmonic frequency related to the rotation periodand show good repeatability during 4 bidirectional rotationperiods In a single bidirectional period the fluctuationson the horizontal accelerometers are also axial symmetryThe fluctuation amplitude is about 150 120583g which means themaximum deviation between the rotation plane and the fixedplane is about 30 arc seconds and it coincides with theamplitude of pitch and roll output fluctuation in Figure 6

The residuals of horizontal accelerometers can be trans-formed into deviation angles between the rotation plane and

Mathematical Problems in Engineering 7

120575120574

120575120574

120575120579

120575120579O

zs

xs

ys

y998400s

x998400s

z998400s

Figure 12 Transformation relation between s frame and 1199041015840 frame

the fixed plane which are defined as 120575120579 120575120574 as shown inFigure 9 (right) and are given as follows

120575120579 =

119886

1015840

119910

119892

120575120574 = minus

119886

1015840

119909

119892

(9)

To compensate influence caused by 120575120579 120575120574 anothercoordinate frame should be defined in the proposed RINS todistinguish actual irregular rotation from ideal rotation Theideal rotation frame is named as 1199041015840 frame whose O-1199111015840

119904

axis iscoinciding with O-zb axis all the time The previous 119904 frameis the actual rotation frame representing the instantaneousrotation frame and it fluctuates around the ideal fixed planeTaking into consideration irregular rotation attitude matrixobtained by (6) should be described as 119862119899

1199041015840 The transforma-

tion matrix from s frame to 1199041015840 frame can be acquired by 120575120579120575120574 and then the output attitude matrix 119862119899

119887

can be updated by

119862

119899

119887

= 119862

119899

1199041015840119862

119904

1015840

119904

119862

119904

119887

(10)

Figure 12 shows the relationship between s frame and 1199041015840

frameThe transformationmatrix1198621199041015840

119904

can be obtained by twoEuler angle rotations according to

119862

119904

1015840

119904

= 119877

119910(120575120574) 119877

119909(120575120579)

(11)

where 119877

119910(120575120574) = [

cos 120575120574 0 minus sin 1205751205740 1 0

sin 120575120574 0 cos 120575120574] and 119877

119909(120575120579) =

[

1 0 00 cos 120575120579 sin 1205751205790 minus sin 120575120579 cos 120575120579

]Since deviation angles 120575120579 120575120574 between 119904 frame and 119904

1015840

frame are small the matrix 1198621199041015840

119904

can be simplified as follows

119862

119904

1015840

119904

asymp

[

[

[

1 0 minus120575120574

0 1 120575120579

120575120574 minus120575120579 1

]

]

]

(12)

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

OriginalFitted

Angle (∘)

Enco

der a

ngle

erro

r (998400998400

)

Figure 13 Relationship between encoder angle error and rotationangle

4 RINS Attitude Error Compensation andExperimental Verification

According to the previous analysis both encoder angle errorsand axis irregular rotation can be corrected and then RINSattitude error fluctuation can be mostly compensated thusimproving short-term attitude output accuracy

41 AzimuthOutput Correction Experimental Results in RINSIt can be seen in Figure 8 that 120575120593 is axial symmetry aboutforward and reverse rotation thus it is closely related toencoder angle 120593 and the relationship between 120575120593 and 120593 isshown in Figure 13

Figure 13 indicates that the fluctuation of 120575120593 presents fun-damental frequencywith respect to the rotation angle Conse-quently the error variation can be compensated through datafitting by the mathematical model given in

120575120593 = 1198960 + 1198961 sin120593+ 1198962 cos120593 (13)

Use 119883 to denote state variable where 119883 = [1198960 1198961 1198962]119879

and then the coefficients 1198960 1198961 1198962 can be acquired throughleast square fitting algorithm by the following measurementsequation

119885 = 119867119883 (14)

where 119885 = [1205751205931 1205751205932 sdot sdot sdot 120575120593

119898]

119879 119867 = [

1 sin1205931 cos12059311 sin1205932 cos1205932sdotsdotsdot sdotsdotsdot sdotsdotsdot

1 sin120593119898

cos120593119898

]

119898times3

and119898 denotes the total number of useful measurementsSince the coefficients 1198960 1198961 1198962 fitting results are acquired

(in this RINS prototype the coefficients fitting results are1198960 = 0032 1198961 = minus6647 and 1198962 = 1249) the encoder angleerror in RINS could be corrected in real time by correctingcoefficients 1198960 1198961 1198962 according to

8 Mathematical Problems in Engineering

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

Angle (∘)

Resid

uals

(998400998400)

Figure 14 Fitting residuals of encoder angle errors

0 500 1000 1500 2000 2500 3000 3500 4000minus120

minus80

minus40

0

40

80

120

160

Time (s)

OriginalCompensated

Azi

mut

h er

ror (

998400998400)

Figure 15 Comparison of original and compensated azimutherrors

120593

119862= 120593minus (1198960 + 1198961 sin120593+ 1198962 cos120593) (15)

where 120593119862denotes the encoder angle to be compensated

Figure 14 shows the fitting residuals of encoder angleerrors and it is less than 2010158401015840 Then the transformation matrix119862

119904

119887

can be calculated by 120593119862instead of 120593 The azimuth output

correction results and comparisons are shown in Figure 15It can be seen from Figure 15 that after azimuth out-

put correction the peak-to-peak azimuth errors decreasedfrom 2sim3 arc minutes to less than 05 arc minutes whichis improved nearly 5 times and it means the short-termazimuth output accuracy achieves significantly improvement

42 Pitch and Roll Compensation Results in RINS The actualexperimental data also show that being similar to encoderangle errors the derivation angles 120575120579 120575120574 acquired by resid-uals of horizontal accelerometers are axial symmetry withrespect to encoder angle 120593 which are shown in Figure 16 and

Table 2 Coefficients estimation results (10158401015840)

1198861 1198871 1198862 1198872

120575120579 minus4209 minus6664 7793 6057120575120574 minus6414 4108 minus0127 minus9708

they can be used to correct and compensate pitch and rolloutput

From Figure 16 it can be seen that the fluctuation of 120575120579120575120574mainly presents second harmonic frequency with respectto rotation period Consequently it can be modeled by (16)and the coefficients estimation results are shown in Table 2Consider

120575120579 = 119886

1205791 sin120593+ 1198871205791 cos120593+ 1198861205792 sin 2120593+ 1198871205792 cos 2120593

120575120574 = 119886

1205741 sin120593+ 1198871205741 cos120593+ 1198861205742 sin 2120593+ 1198871205742 cos 2120593(16)

Figure 17 shows the fitting residuals of 120575120579 120575120574 and theyare less than 510158401015840 Then the transformation matrix 119862119904

119887

can becalculated according to (10) and (11) Thus the pitch and rolloutput compensation results can be acquired as shown inFigure 18

Conclusions can be drawn from Figure 18 that after pitchand roll output compensation the peak-to-peak pitch and rollerrors decreased from 20sim30 arc seconds to less than 5 arcseconds which is improved nearly 5 times It is proved thatthe proposed output compensation algorithm for pitch androll is valid and the pitch and roll short-term output accuracyis improved obviously

However it is worth mentioning that the compensationalgorithm here is not only suitable for single-axis rotationRINS For dual-axis and tri-axis RINS it is inevitable tohave encoder installation eccentricity and other mentionedproblems meanwhile the irregular rotation is hard to avoidas well consequently the analysis and attitude compensationand correction algorithm presented in this paper are alsosuitable for other types of RINS

5 Conclusion

This paper researched the attitude output accuracy improve-ment in rotation RINS Comparative experiment resultsindicate that velocity and position accuracy gains greatimprovement in rotation mode while attitude accuracy iseven worse than that in strapdown mode The reasons thatcause attitude output accuracy loss are analyzed and thena new attitude output compensation algorithm for RINS ispresented The experimental results proved validity of theproposed attitude correction and compensation algorithmwith short-term pitch and roll output accuracy improvedfrom 20sim30 arc seconds to less than 5 arc seconds andazimuth output accuracy improved from 2sim3 arc minutes toless than 05 arc minutes

According to dead reckoning principle high accuracy invelocity and position should be matched with high attitudeaccuracy The proposed compensation algorithm in thispaper solved the problem of attitude accuracy loss of RINSin practical applications which is significant for many task

Mathematical Problems in Engineering 9

0 60 120 180 240 300 360minus20

0

20

40

OriginalFitted

OriginalFitted

0 60 120 180 240 300 360minus20

0

20

40

120575120574

(998400998400)

120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 16 Relationships between 120575120579 120575120574 and rotation angle

0 60 120 180 240 300 360minus20

0

20

40

0 60 120 180 240 300 360minus20

0

20

40

Resid

uals

of120575120574

(998400998400)

Resid

uals

of120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 17 Fitting residuals of 120575120579 120575120574

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

OriginalCompensated

OriginalCompensated

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

Pitc

h er

ror (

998400998400)

Roll

erro

r (998400998400

)

Figure 18 Comparison of original and compensated pitch and roll errors

systems where attitude accuracy is urgently required Inaddition the proposed compensation algorithm is a generalmethod it is not only suitable for single-axis RINS but it canalso be used to improve short-term attitude output accuracyin dual-axis and tri-axis RINS

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported and funded by the National Nat-ural Science Foundation of China (L142200032) and Long-Term Development Strategic Research of China EngineeringScience and Technology (2014-zcq-01) The authors appreci-ate the support and fund

References

[1] E S Geller ldquoInertial system platform rotationrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol AES-4 no 4 pp557ndash568 1968

[2] E Levinson and R Majure ldquoAccuracy enhancement techniquesapplied to the marine ring laser inertial navigator (MARLIN)rdquoNavigation vol 34 no 1 pp 64ndash86 1987

[3] E Levinson J ter Horst and M Willcocks ldquoThe next genera-tion marine inertial navigator is here nowrdquo in Proceedings of thePosition Location and Navigation Symposium pp 121ndash127 IEEEApril 1994

[4] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation Journal ofthe Institute of Navigation vol 28 no 4 pp 311ndash341 1981

[5] Y Yang and L-J Miao ldquoFiber-optic strapdown inertial systemwith sensing cluster continuous rotationrdquo IEEE Transactions onAerospace and Electronic Systems vol 40 no 4 pp 1173ndash11782004

10 Mathematical Problems in Engineering

[6] S Ishibashi S Tsukioka H Yoshida et al ldquoAccuracy improve-ment of an inertial navigation system brought about by therotational motionrdquo in Proceedings of the OCEANSmdashEurope pp1ndash5 IEEE Aberdeen Scotland June 2007

[7] S Ishibashi S Tsukioka T Sawa et al ldquoThe rotation controlsystem to improve the accuracy of an inertial navigation systeminstalled in an autonomous underwater vehiclerdquo in Proceedingsof the Symposium on Underwater Technology and Workshop onScientific Use of Submarine Cables and Related Technologies pp495ndash498 IEEE Tokyo Japan April 2007

[8] A Li G-B Chang F-J Qin and H-W Li ldquoImproved precisionof strapdown inertial navigation system brought by dual-axiscontinuous rotation of inertial measurement unitrdquo in Proceed-ings of the 2nd International Asia Conference on Informatics inControl Automation and Robotics (CAR rsquo10) vol 1 pp 284ndash287March 2010

[9] G Chang J Xu A Li and K Cheng ldquoError analysis andsimulation of the dual-axis rotation-dwell autocompensatingstrapdown inertial navigation systemrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo10) pp 124ndash127 March 2010

[10] Z Deng M Sun and B Wang ldquoError modulation schemeanalysis of dual-axis rotating strap-down inertial navigationsystem based on FOGrdquo in Proceedings of the 33rd ChineseControl Conference (CCC rsquo14) pp 692ndash696Nanjing China July2014

[11] C Jianhua LMingyue CDaidai C Li and S Junyu ldquoResearchof strapdown inertial navigation system monitor techniquebased on dual-axis consequential rotationrdquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 203ndash208 June 2011

[12] R B Morrow Jr and D W Heckman ldquoHigh precision IFOGinsertion into the strategic submarine navigation systemrdquo inProceedings of the IEEE Position Location and Navigation Sym-posium pp 332ndash338 IEEE Palm Springs Calif USA April1998

[13] D W Heckman and L M Baretela ldquoImproved affordabilityof high precision submarine inertial navigation by insertion ofrapidly developing fiber optic gyro technologyrdquo in Proceedingsof the IEEE Position Location and and Navigation Symposiumpp 404ndash410 March 2000

[14] S Qin Z Huang and X Wang ldquoOptical angular encoderinstallation error measurement and calibration by ring lasergyroscoperdquo IEEETransactions on Instrumentation andMeasure-ment vol 59 no 3 pp 506ndash511 2010

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

minus2000

minus1000

0

1000

Time (s)

PE er

ror (

m)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 40000 500 1000 1500 2000 2500 3000 3500 4000minus500

0

500

1000

Time (s)

PN er

ror (

m)

RotationStrapdown

Figure 4 PE and PN errors comparison

0 500 1000 1500 2000 2500 3000 3500 4000minus2

minus1

0

1

Time (s)

VE

erro

r (m

s)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 4000minus05

0

05

1

Time (s)

VN

erro

r (m

s)

RotationStrapdown

Figure 5 VE and VN errors comparison

0 500 1000 1500 2000 2500 3000 3500 4000minus50

0

50

Time (s)

RotationStrapdown

Pitc

h er

ror (

998400998400)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 4000minus50

0

50

Time (s)

Roll

erro

r (998400998400

)

RotationStrapdown

0 500 1000 1500 2000 2500 3000 3500 4000minus200

0

200

Time (s)

Azi

mut

h er

ror (

998400998400)

Figure 6 Strapdown and rotation INS attitude errors comparison

strapdown INS configuration 119862119899119887

is calculated directly fromrotation vector by quaternion attitude updating algorithmHowever in RINS configuration attitude matrix 119862119899

119904

shouldbe updated beforehand (the update process ofmatrix119862119899

119904

is thesame as that of119862119899

119887

in strapdown INS) then119862119899119887

can be acquiredaccording to the following equation

119862

119899

119887

= 119862

119899

119904

119862

119904

119887

(4)

The matrix 119862119904119887

is the transformation matrix from 119887 frameto 119904 frame and is given as follows

119862

119904

119887

=

[

[

[

cos120593 sin120593 0minus sin120593 cos120593 0

0 0 1

]

]

]

(5)

where 120593 is the rotation angle provided by encoder

Mathematical Problems in Engineering 5

If all the updating processes above are ideal attitudeoutput in RINS should present high accuracy and shouldnot contain short-term error fluctuation As a matter of factin RINS updating matrix 119862

119899

119904

is calculated from rotationvector by quaternion attitude updating algorithm which isthe same as 119862119899

119887

in strapdown INS thus it is accurate enoughConsequently the most possible cause of RINS short-termattitude output accuracy loss is transformation process fromattitude matrix 119862119899

119904

to 119862119899119887

that is to say matrix 119862119904119887

In actualRINS the encoder output errors installation eccentricity ofencoder and noncoaxial rotation of IMU can cause rotationangle 120593 containing periodical output angle errors [14] whichlead to azimuth output fluctuation inRINS Besides if the axisO-zs does not coincidewith the rotation axisO-zb all the timethe rotation angular rate will have projection along horizontalaxes O-xs and O-ys Thus undesirable influences will beattached to the transformation matrix 119862119904

119887

which will resultin error fluctuation on pitch and roll output The detailedanalysis will be given as follows

32 Analysis of Azimuth Output Error in RINS Equation (5)indicates that the rotation angle 120593 will have effect on theaccuracy of matrix 119862119904

119887

Therefore it is necessary to analyzethe rotation angle provided by the encoder first In orderto obtain the encoder angle errors the angle integrated byoutput angular rate of gyroscope 119885 is used However outputof gyroscope 119885 contains not only rotation angular rate 120596 butalso gyroscope drift 120576

119911and the upward component of earth

rotation angular rate 120596119894119890sin 119871 The sign of IMU rotation rate

120596 is opposite during forward and reverse rotation Then thegyroscope outputs are given as

120596

119911119891= 120596+120596

119894119890sin 119871+ 120576

119911

120596

119911119903= minus120596+120596

119894119890sin 119871+ 120576

119911

(6)

where subscript 119891 denotes forward rotation and 119903 denotesreverse rotation If we define 1205961015840 as 120596

119894119890sin 119871 + 120576

119911= (120596

119911119891+

120596

119911119903)2 then the angle integrated by output angular rate

of gyroscope 119885 during IMU rotation can be calculatedaccording to (7) which will be a reference for encoder angleoutput Consider

120593

119911119891= int (120596

119911119891minus120596

1015840

) 119889119905

120593

119911119903= int (120596

119911119903minus120596

1015840

) 119889119905

(7)

Figure 7 shows actual experimental data of encoder angleoutputs and Figure 8 is acquired when subtracting encoderangle from the integrated angle of gyroscope 119885 It can beseen from the two figures that encoder angle error (defineas 120575120593) presents obvious repeatability during 4 bidirectionalrotation periods and the fluctuations are axial symmetryabout forward and reverse rotation in a separate bidirectionalperiod 120575120593 presents periodical fluctuation with the sameperiod as IMU rotation and the fluctuation amplitude is 1sim2 arc minutes It presents the same property as azimuth errorin Figure 6 and can be compensated afterwards short-termazimuth output accuracy would be greatly improved

0 500 1000 1500 2000 2500 3000 3500 40000

50

100

150

200

250

300

350

400

Time (s)

Enco

der o

utpu

t (∘ )

Figure 7 Encoder angular output

0 500 1000 1500 2000 2500 3000 3500 4000minus150

minus100

minus50

0

50

100

150

Time (s)

Enco

der a

ngle

erro

r (998400998400

)

Figure 8 Encoder angular error

33 Analysis of Pitch and Roll Output Error in RINS Theexpression of matrix 119862

119904

119887

in (5) is obtained based on idealrotation namely the axes of O-xs and O-ys stay in a fixedplane during rotation However in fact irregular rotation ishard to avoid and the rotation planewould change in differentpositions The difference between actual rotation and idealrotation should be analyzed in detail

Figure 9 (left) describes IMU rotation under ideal condi-tion and in this case axes of O-xs and O-ys in 119904 frame remainunchanged in fixed plane and axis O-zs coincides with O-zbin body frame all the time However in actual condition dueto some reasons such as machining and assembling errors inrotation axis and defects in rotation bearings the IMU rotatesirregularly Figure 9 (right) shows this irregular rotation andthe rotation plane in 119904 frame no longer stays unchanged thepointing of O-zs axis will change in space and axes of O-xsand O-ys will fluctuate around the ideal fixed plane

If the rotation plane remains unchanged the projectionof acceleration of gravity on horizontal accelerometers will

6 Mathematical Problems in Engineering

Actual rotation axis

Rotation axis

ys

Ay

Az

xs

xs

AxAx

zszb

ysAz

Ay 120575120579

120575120574

zb(zs)

Figure 9 Comparison of ideal rotation and actual irregular rotation

0 500 1000 1500 2000 2500 3000 3500 4000minus004

minus002

0

002

004

Time (s)0 500 1000 1500 2000 2500 3000 3500 4000

minus004

minus002

0

002

004

Time (s)

Acce

lero

met

er x

(ms2)

Acce

lero

met

er y

(ms2)

Figure 10 Initial outputs of horizontal accelerometers

0 500 1000 1500 2000 2500 3000 3500 4000minus200

minus100

0

100

200

Time (s)0 500 1000 1500 2000 2500 3000 3500 4000

minus200

minus100

0

100

200

Time (s)

Resid

ual o

f acc

elero

met

er120583

g)

Resid

ual o

f acc

eler

omet

er120583

g)

x (

y (

Figure 11 Residual components of horizontal accelerometers

be standard sine or cosine form with the same period asIMU rotation and the amplitude is determined by initialpitch and roll of the rotation plane However due to thefluctuation of rotation plane in actual situation the projectionof acceleration of gravity on horizontal accelerometers willbring about irregular variation and the deviation betweenthe rotation plane and the fixed plane can be representedby irregular variation in horizontal accelerometers Figure 10shows the original output of two horizontal accelerometers inactual experiments In order to obtain the irregular rotationcomponents firstly the projection of acceleration of gravitycaused by pitch and roll should be deducted from thehorizontal accelerometers according to

119886

1015840

119909

= 119886

119909+1198921205740 cos120593minus1198921205790 sin120593

119886

1015840

119910

= 119886

119910minus1198921205790 cos120593minus1198921205740 sin120593

(8)

where 1205790 1205740 are initial pitch and roll acquired by thealignment process in RINS

Figure 11 shows the residual components of horizon-tal accelerometers during the experiment Both horizon-tal accelerometers residual components mainly representsecond harmonic frequency related to the rotation periodand show good repeatability during 4 bidirectional rotationperiods In a single bidirectional period the fluctuationson the horizontal accelerometers are also axial symmetryThe fluctuation amplitude is about 150 120583g which means themaximum deviation between the rotation plane and the fixedplane is about 30 arc seconds and it coincides with theamplitude of pitch and roll output fluctuation in Figure 6

The residuals of horizontal accelerometers can be trans-formed into deviation angles between the rotation plane and

Mathematical Problems in Engineering 7

120575120574

120575120574

120575120579

120575120579O

zs

xs

ys

y998400s

x998400s

z998400s

Figure 12 Transformation relation between s frame and 1199041015840 frame

the fixed plane which are defined as 120575120579 120575120574 as shown inFigure 9 (right) and are given as follows

120575120579 =

119886

1015840

119910

119892

120575120574 = minus

119886

1015840

119909

119892

(9)

To compensate influence caused by 120575120579 120575120574 anothercoordinate frame should be defined in the proposed RINS todistinguish actual irregular rotation from ideal rotation Theideal rotation frame is named as 1199041015840 frame whose O-1199111015840

119904

axis iscoinciding with O-zb axis all the time The previous 119904 frameis the actual rotation frame representing the instantaneousrotation frame and it fluctuates around the ideal fixed planeTaking into consideration irregular rotation attitude matrixobtained by (6) should be described as 119862119899

1199041015840 The transforma-

tion matrix from s frame to 1199041015840 frame can be acquired by 120575120579120575120574 and then the output attitude matrix 119862119899

119887

can be updated by

119862

119899

119887

= 119862

119899

1199041015840119862

119904

1015840

119904

119862

119904

119887

(10)

Figure 12 shows the relationship between s frame and 1199041015840

frameThe transformationmatrix1198621199041015840

119904

can be obtained by twoEuler angle rotations according to

119862

119904

1015840

119904

= 119877

119910(120575120574) 119877

119909(120575120579)

(11)

where 119877

119910(120575120574) = [

cos 120575120574 0 minus sin 1205751205740 1 0

sin 120575120574 0 cos 120575120574] and 119877

119909(120575120579) =

[

1 0 00 cos 120575120579 sin 1205751205790 minus sin 120575120579 cos 120575120579

]Since deviation angles 120575120579 120575120574 between 119904 frame and 119904

1015840

frame are small the matrix 1198621199041015840

119904

can be simplified as follows

119862

119904

1015840

119904

asymp

[

[

[

1 0 minus120575120574

0 1 120575120579

120575120574 minus120575120579 1

]

]

]

(12)

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

OriginalFitted

Angle (∘)

Enco

der a

ngle

erro

r (998400998400

)

Figure 13 Relationship between encoder angle error and rotationangle

4 RINS Attitude Error Compensation andExperimental Verification

According to the previous analysis both encoder angle errorsand axis irregular rotation can be corrected and then RINSattitude error fluctuation can be mostly compensated thusimproving short-term attitude output accuracy

41 AzimuthOutput Correction Experimental Results in RINSIt can be seen in Figure 8 that 120575120593 is axial symmetry aboutforward and reverse rotation thus it is closely related toencoder angle 120593 and the relationship between 120575120593 and 120593 isshown in Figure 13

Figure 13 indicates that the fluctuation of 120575120593 presents fun-damental frequencywith respect to the rotation angle Conse-quently the error variation can be compensated through datafitting by the mathematical model given in

120575120593 = 1198960 + 1198961 sin120593+ 1198962 cos120593 (13)

Use 119883 to denote state variable where 119883 = [1198960 1198961 1198962]119879

and then the coefficients 1198960 1198961 1198962 can be acquired throughleast square fitting algorithm by the following measurementsequation

119885 = 119867119883 (14)

where 119885 = [1205751205931 1205751205932 sdot sdot sdot 120575120593

119898]

119879 119867 = [

1 sin1205931 cos12059311 sin1205932 cos1205932sdotsdotsdot sdotsdotsdot sdotsdotsdot

1 sin120593119898

cos120593119898

]

119898times3

and119898 denotes the total number of useful measurementsSince the coefficients 1198960 1198961 1198962 fitting results are acquired

(in this RINS prototype the coefficients fitting results are1198960 = 0032 1198961 = minus6647 and 1198962 = 1249) the encoder angleerror in RINS could be corrected in real time by correctingcoefficients 1198960 1198961 1198962 according to

8 Mathematical Problems in Engineering

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

Angle (∘)

Resid

uals

(998400998400)

Figure 14 Fitting residuals of encoder angle errors

0 500 1000 1500 2000 2500 3000 3500 4000minus120

minus80

minus40

0

40

80

120

160

Time (s)

OriginalCompensated

Azi

mut

h er

ror (

998400998400)

Figure 15 Comparison of original and compensated azimutherrors

120593

119862= 120593minus (1198960 + 1198961 sin120593+ 1198962 cos120593) (15)

where 120593119862denotes the encoder angle to be compensated

Figure 14 shows the fitting residuals of encoder angleerrors and it is less than 2010158401015840 Then the transformation matrix119862

119904

119887

can be calculated by 120593119862instead of 120593 The azimuth output

correction results and comparisons are shown in Figure 15It can be seen from Figure 15 that after azimuth out-

put correction the peak-to-peak azimuth errors decreasedfrom 2sim3 arc minutes to less than 05 arc minutes whichis improved nearly 5 times and it means the short-termazimuth output accuracy achieves significantly improvement

42 Pitch and Roll Compensation Results in RINS The actualexperimental data also show that being similar to encoderangle errors the derivation angles 120575120579 120575120574 acquired by resid-uals of horizontal accelerometers are axial symmetry withrespect to encoder angle 120593 which are shown in Figure 16 and

Table 2 Coefficients estimation results (10158401015840)

1198861 1198871 1198862 1198872

120575120579 minus4209 minus6664 7793 6057120575120574 minus6414 4108 minus0127 minus9708

they can be used to correct and compensate pitch and rolloutput

From Figure 16 it can be seen that the fluctuation of 120575120579120575120574mainly presents second harmonic frequency with respectto rotation period Consequently it can be modeled by (16)and the coefficients estimation results are shown in Table 2Consider

120575120579 = 119886

1205791 sin120593+ 1198871205791 cos120593+ 1198861205792 sin 2120593+ 1198871205792 cos 2120593

120575120574 = 119886

1205741 sin120593+ 1198871205741 cos120593+ 1198861205742 sin 2120593+ 1198871205742 cos 2120593(16)

Figure 17 shows the fitting residuals of 120575120579 120575120574 and theyare less than 510158401015840 Then the transformation matrix 119862119904

119887

can becalculated according to (10) and (11) Thus the pitch and rolloutput compensation results can be acquired as shown inFigure 18

Conclusions can be drawn from Figure 18 that after pitchand roll output compensation the peak-to-peak pitch and rollerrors decreased from 20sim30 arc seconds to less than 5 arcseconds which is improved nearly 5 times It is proved thatthe proposed output compensation algorithm for pitch androll is valid and the pitch and roll short-term output accuracyis improved obviously

However it is worth mentioning that the compensationalgorithm here is not only suitable for single-axis rotationRINS For dual-axis and tri-axis RINS it is inevitable tohave encoder installation eccentricity and other mentionedproblems meanwhile the irregular rotation is hard to avoidas well consequently the analysis and attitude compensationand correction algorithm presented in this paper are alsosuitable for other types of RINS

5 Conclusion

This paper researched the attitude output accuracy improve-ment in rotation RINS Comparative experiment resultsindicate that velocity and position accuracy gains greatimprovement in rotation mode while attitude accuracy iseven worse than that in strapdown mode The reasons thatcause attitude output accuracy loss are analyzed and thena new attitude output compensation algorithm for RINS ispresented The experimental results proved validity of theproposed attitude correction and compensation algorithmwith short-term pitch and roll output accuracy improvedfrom 20sim30 arc seconds to less than 5 arc seconds andazimuth output accuracy improved from 2sim3 arc minutes toless than 05 arc minutes

According to dead reckoning principle high accuracy invelocity and position should be matched with high attitudeaccuracy The proposed compensation algorithm in thispaper solved the problem of attitude accuracy loss of RINSin practical applications which is significant for many task

Mathematical Problems in Engineering 9

0 60 120 180 240 300 360minus20

0

20

40

OriginalFitted

OriginalFitted

0 60 120 180 240 300 360minus20

0

20

40

120575120574

(998400998400)

120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 16 Relationships between 120575120579 120575120574 and rotation angle

0 60 120 180 240 300 360minus20

0

20

40

0 60 120 180 240 300 360minus20

0

20

40

Resid

uals

of120575120574

(998400998400)

Resid

uals

of120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 17 Fitting residuals of 120575120579 120575120574

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

OriginalCompensated

OriginalCompensated

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

Pitc

h er

ror (

998400998400)

Roll

erro

r (998400998400

)

Figure 18 Comparison of original and compensated pitch and roll errors

systems where attitude accuracy is urgently required Inaddition the proposed compensation algorithm is a generalmethod it is not only suitable for single-axis RINS but it canalso be used to improve short-term attitude output accuracyin dual-axis and tri-axis RINS

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported and funded by the National Nat-ural Science Foundation of China (L142200032) and Long-Term Development Strategic Research of China EngineeringScience and Technology (2014-zcq-01) The authors appreci-ate the support and fund

References

[1] E S Geller ldquoInertial system platform rotationrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol AES-4 no 4 pp557ndash568 1968

[2] E Levinson and R Majure ldquoAccuracy enhancement techniquesapplied to the marine ring laser inertial navigator (MARLIN)rdquoNavigation vol 34 no 1 pp 64ndash86 1987

[3] E Levinson J ter Horst and M Willcocks ldquoThe next genera-tion marine inertial navigator is here nowrdquo in Proceedings of thePosition Location and Navigation Symposium pp 121ndash127 IEEEApril 1994

[4] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation Journal ofthe Institute of Navigation vol 28 no 4 pp 311ndash341 1981

[5] Y Yang and L-J Miao ldquoFiber-optic strapdown inertial systemwith sensing cluster continuous rotationrdquo IEEE Transactions onAerospace and Electronic Systems vol 40 no 4 pp 1173ndash11782004

10 Mathematical Problems in Engineering

[6] S Ishibashi S Tsukioka H Yoshida et al ldquoAccuracy improve-ment of an inertial navigation system brought about by therotational motionrdquo in Proceedings of the OCEANSmdashEurope pp1ndash5 IEEE Aberdeen Scotland June 2007

[7] S Ishibashi S Tsukioka T Sawa et al ldquoThe rotation controlsystem to improve the accuracy of an inertial navigation systeminstalled in an autonomous underwater vehiclerdquo in Proceedingsof the Symposium on Underwater Technology and Workshop onScientific Use of Submarine Cables and Related Technologies pp495ndash498 IEEE Tokyo Japan April 2007

[8] A Li G-B Chang F-J Qin and H-W Li ldquoImproved precisionof strapdown inertial navigation system brought by dual-axiscontinuous rotation of inertial measurement unitrdquo in Proceed-ings of the 2nd International Asia Conference on Informatics inControl Automation and Robotics (CAR rsquo10) vol 1 pp 284ndash287March 2010

[9] G Chang J Xu A Li and K Cheng ldquoError analysis andsimulation of the dual-axis rotation-dwell autocompensatingstrapdown inertial navigation systemrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo10) pp 124ndash127 March 2010

[10] Z Deng M Sun and B Wang ldquoError modulation schemeanalysis of dual-axis rotating strap-down inertial navigationsystem based on FOGrdquo in Proceedings of the 33rd ChineseControl Conference (CCC rsquo14) pp 692ndash696Nanjing China July2014

[11] C Jianhua LMingyue CDaidai C Li and S Junyu ldquoResearchof strapdown inertial navigation system monitor techniquebased on dual-axis consequential rotationrdquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 203ndash208 June 2011

[12] R B Morrow Jr and D W Heckman ldquoHigh precision IFOGinsertion into the strategic submarine navigation systemrdquo inProceedings of the IEEE Position Location and Navigation Sym-posium pp 332ndash338 IEEE Palm Springs Calif USA April1998

[13] D W Heckman and L M Baretela ldquoImproved affordabilityof high precision submarine inertial navigation by insertion ofrapidly developing fiber optic gyro technologyrdquo in Proceedingsof the IEEE Position Location and and Navigation Symposiumpp 404ndash410 March 2000

[14] S Qin Z Huang and X Wang ldquoOptical angular encoderinstallation error measurement and calibration by ring lasergyroscoperdquo IEEETransactions on Instrumentation andMeasure-ment vol 59 no 3 pp 506ndash511 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

If all the updating processes above are ideal attitudeoutput in RINS should present high accuracy and shouldnot contain short-term error fluctuation As a matter of factin RINS updating matrix 119862

119899

119904

is calculated from rotationvector by quaternion attitude updating algorithm which isthe same as 119862119899

119887

in strapdown INS thus it is accurate enoughConsequently the most possible cause of RINS short-termattitude output accuracy loss is transformation process fromattitude matrix 119862119899

119904

to 119862119899119887

that is to say matrix 119862119904119887

In actualRINS the encoder output errors installation eccentricity ofencoder and noncoaxial rotation of IMU can cause rotationangle 120593 containing periodical output angle errors [14] whichlead to azimuth output fluctuation inRINS Besides if the axisO-zs does not coincidewith the rotation axisO-zb all the timethe rotation angular rate will have projection along horizontalaxes O-xs and O-ys Thus undesirable influences will beattached to the transformation matrix 119862119904

119887

which will resultin error fluctuation on pitch and roll output The detailedanalysis will be given as follows

32 Analysis of Azimuth Output Error in RINS Equation (5)indicates that the rotation angle 120593 will have effect on theaccuracy of matrix 119862119904

119887

Therefore it is necessary to analyzethe rotation angle provided by the encoder first In orderto obtain the encoder angle errors the angle integrated byoutput angular rate of gyroscope 119885 is used However outputof gyroscope 119885 contains not only rotation angular rate 120596 butalso gyroscope drift 120576

119911and the upward component of earth

rotation angular rate 120596119894119890sin 119871 The sign of IMU rotation rate

120596 is opposite during forward and reverse rotation Then thegyroscope outputs are given as

120596

119911119891= 120596+120596

119894119890sin 119871+ 120576

119911

120596

119911119903= minus120596+120596

119894119890sin 119871+ 120576

119911

(6)

where subscript 119891 denotes forward rotation and 119903 denotesreverse rotation If we define 1205961015840 as 120596

119894119890sin 119871 + 120576

119911= (120596

119911119891+

120596

119911119903)2 then the angle integrated by output angular rate

of gyroscope 119885 during IMU rotation can be calculatedaccording to (7) which will be a reference for encoder angleoutput Consider

120593

119911119891= int (120596

119911119891minus120596

1015840

) 119889119905

120593

119911119903= int (120596

119911119903minus120596

1015840

) 119889119905

(7)

Figure 7 shows actual experimental data of encoder angleoutputs and Figure 8 is acquired when subtracting encoderangle from the integrated angle of gyroscope 119885 It can beseen from the two figures that encoder angle error (defineas 120575120593) presents obvious repeatability during 4 bidirectionalrotation periods and the fluctuations are axial symmetryabout forward and reverse rotation in a separate bidirectionalperiod 120575120593 presents periodical fluctuation with the sameperiod as IMU rotation and the fluctuation amplitude is 1sim2 arc minutes It presents the same property as azimuth errorin Figure 6 and can be compensated afterwards short-termazimuth output accuracy would be greatly improved

0 500 1000 1500 2000 2500 3000 3500 40000

50

100

150

200

250

300

350

400

Time (s)

Enco

der o

utpu

t (∘ )

Figure 7 Encoder angular output

0 500 1000 1500 2000 2500 3000 3500 4000minus150

minus100

minus50

0

50

100

150

Time (s)

Enco

der a

ngle

erro

r (998400998400

)

Figure 8 Encoder angular error

33 Analysis of Pitch and Roll Output Error in RINS Theexpression of matrix 119862

119904

119887

in (5) is obtained based on idealrotation namely the axes of O-xs and O-ys stay in a fixedplane during rotation However in fact irregular rotation ishard to avoid and the rotation planewould change in differentpositions The difference between actual rotation and idealrotation should be analyzed in detail

Figure 9 (left) describes IMU rotation under ideal condi-tion and in this case axes of O-xs and O-ys in 119904 frame remainunchanged in fixed plane and axis O-zs coincides with O-zbin body frame all the time However in actual condition dueto some reasons such as machining and assembling errors inrotation axis and defects in rotation bearings the IMU rotatesirregularly Figure 9 (right) shows this irregular rotation andthe rotation plane in 119904 frame no longer stays unchanged thepointing of O-zs axis will change in space and axes of O-xsand O-ys will fluctuate around the ideal fixed plane

If the rotation plane remains unchanged the projectionof acceleration of gravity on horizontal accelerometers will

6 Mathematical Problems in Engineering

Actual rotation axis

Rotation axis

ys

Ay

Az

xs

xs

AxAx

zszb

ysAz

Ay 120575120579

120575120574

zb(zs)

Figure 9 Comparison of ideal rotation and actual irregular rotation

0 500 1000 1500 2000 2500 3000 3500 4000minus004

minus002

0

002

004

Time (s)0 500 1000 1500 2000 2500 3000 3500 4000

minus004

minus002

0

002

004

Time (s)

Acce

lero

met

er x

(ms2)

Acce

lero

met

er y

(ms2)

Figure 10 Initial outputs of horizontal accelerometers

0 500 1000 1500 2000 2500 3000 3500 4000minus200

minus100

0

100

200

Time (s)0 500 1000 1500 2000 2500 3000 3500 4000

minus200

minus100

0

100

200

Time (s)

Resid

ual o

f acc

elero

met

er120583

g)

Resid

ual o

f acc

eler

omet

er120583

g)

x (

y (

Figure 11 Residual components of horizontal accelerometers

be standard sine or cosine form with the same period asIMU rotation and the amplitude is determined by initialpitch and roll of the rotation plane However due to thefluctuation of rotation plane in actual situation the projectionof acceleration of gravity on horizontal accelerometers willbring about irregular variation and the deviation betweenthe rotation plane and the fixed plane can be representedby irregular variation in horizontal accelerometers Figure 10shows the original output of two horizontal accelerometers inactual experiments In order to obtain the irregular rotationcomponents firstly the projection of acceleration of gravitycaused by pitch and roll should be deducted from thehorizontal accelerometers according to

119886

1015840

119909

= 119886

119909+1198921205740 cos120593minus1198921205790 sin120593

119886

1015840

119910

= 119886

119910minus1198921205790 cos120593minus1198921205740 sin120593

(8)

where 1205790 1205740 are initial pitch and roll acquired by thealignment process in RINS

Figure 11 shows the residual components of horizon-tal accelerometers during the experiment Both horizon-tal accelerometers residual components mainly representsecond harmonic frequency related to the rotation periodand show good repeatability during 4 bidirectional rotationperiods In a single bidirectional period the fluctuationson the horizontal accelerometers are also axial symmetryThe fluctuation amplitude is about 150 120583g which means themaximum deviation between the rotation plane and the fixedplane is about 30 arc seconds and it coincides with theamplitude of pitch and roll output fluctuation in Figure 6

The residuals of horizontal accelerometers can be trans-formed into deviation angles between the rotation plane and

Mathematical Problems in Engineering 7

120575120574

120575120574

120575120579

120575120579O

zs

xs

ys

y998400s

x998400s

z998400s

Figure 12 Transformation relation between s frame and 1199041015840 frame

the fixed plane which are defined as 120575120579 120575120574 as shown inFigure 9 (right) and are given as follows

120575120579 =

119886

1015840

119910

119892

120575120574 = minus

119886

1015840

119909

119892

(9)

To compensate influence caused by 120575120579 120575120574 anothercoordinate frame should be defined in the proposed RINS todistinguish actual irregular rotation from ideal rotation Theideal rotation frame is named as 1199041015840 frame whose O-1199111015840

119904

axis iscoinciding with O-zb axis all the time The previous 119904 frameis the actual rotation frame representing the instantaneousrotation frame and it fluctuates around the ideal fixed planeTaking into consideration irregular rotation attitude matrixobtained by (6) should be described as 119862119899

1199041015840 The transforma-

tion matrix from s frame to 1199041015840 frame can be acquired by 120575120579120575120574 and then the output attitude matrix 119862119899

119887

can be updated by

119862

119899

119887

= 119862

119899

1199041015840119862

119904

1015840

119904

119862

119904

119887

(10)

Figure 12 shows the relationship between s frame and 1199041015840

frameThe transformationmatrix1198621199041015840

119904

can be obtained by twoEuler angle rotations according to

119862

119904

1015840

119904

= 119877

119910(120575120574) 119877

119909(120575120579)

(11)

where 119877

119910(120575120574) = [

cos 120575120574 0 minus sin 1205751205740 1 0

sin 120575120574 0 cos 120575120574] and 119877

119909(120575120579) =

[

1 0 00 cos 120575120579 sin 1205751205790 minus sin 120575120579 cos 120575120579

]Since deviation angles 120575120579 120575120574 between 119904 frame and 119904

1015840

frame are small the matrix 1198621199041015840

119904

can be simplified as follows

119862

119904

1015840

119904

asymp

[

[

[

1 0 minus120575120574

0 1 120575120579

120575120574 minus120575120579 1

]

]

]

(12)

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

OriginalFitted

Angle (∘)

Enco

der a

ngle

erro

r (998400998400

)

Figure 13 Relationship between encoder angle error and rotationangle

4 RINS Attitude Error Compensation andExperimental Verification

According to the previous analysis both encoder angle errorsand axis irregular rotation can be corrected and then RINSattitude error fluctuation can be mostly compensated thusimproving short-term attitude output accuracy

41 AzimuthOutput Correction Experimental Results in RINSIt can be seen in Figure 8 that 120575120593 is axial symmetry aboutforward and reverse rotation thus it is closely related toencoder angle 120593 and the relationship between 120575120593 and 120593 isshown in Figure 13

Figure 13 indicates that the fluctuation of 120575120593 presents fun-damental frequencywith respect to the rotation angle Conse-quently the error variation can be compensated through datafitting by the mathematical model given in

120575120593 = 1198960 + 1198961 sin120593+ 1198962 cos120593 (13)

Use 119883 to denote state variable where 119883 = [1198960 1198961 1198962]119879

and then the coefficients 1198960 1198961 1198962 can be acquired throughleast square fitting algorithm by the following measurementsequation

119885 = 119867119883 (14)

where 119885 = [1205751205931 1205751205932 sdot sdot sdot 120575120593

119898]

119879 119867 = [

1 sin1205931 cos12059311 sin1205932 cos1205932sdotsdotsdot sdotsdotsdot sdotsdotsdot

1 sin120593119898

cos120593119898

]

119898times3

and119898 denotes the total number of useful measurementsSince the coefficients 1198960 1198961 1198962 fitting results are acquired

(in this RINS prototype the coefficients fitting results are1198960 = 0032 1198961 = minus6647 and 1198962 = 1249) the encoder angleerror in RINS could be corrected in real time by correctingcoefficients 1198960 1198961 1198962 according to

8 Mathematical Problems in Engineering

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

Angle (∘)

Resid

uals

(998400998400)

Figure 14 Fitting residuals of encoder angle errors

0 500 1000 1500 2000 2500 3000 3500 4000minus120

minus80

minus40

0

40

80

120

160

Time (s)

OriginalCompensated

Azi

mut

h er

ror (

998400998400)

Figure 15 Comparison of original and compensated azimutherrors

120593

119862= 120593minus (1198960 + 1198961 sin120593+ 1198962 cos120593) (15)

where 120593119862denotes the encoder angle to be compensated

Figure 14 shows the fitting residuals of encoder angleerrors and it is less than 2010158401015840 Then the transformation matrix119862

119904

119887

can be calculated by 120593119862instead of 120593 The azimuth output

correction results and comparisons are shown in Figure 15It can be seen from Figure 15 that after azimuth out-

put correction the peak-to-peak azimuth errors decreasedfrom 2sim3 arc minutes to less than 05 arc minutes whichis improved nearly 5 times and it means the short-termazimuth output accuracy achieves significantly improvement

42 Pitch and Roll Compensation Results in RINS The actualexperimental data also show that being similar to encoderangle errors the derivation angles 120575120579 120575120574 acquired by resid-uals of horizontal accelerometers are axial symmetry withrespect to encoder angle 120593 which are shown in Figure 16 and

Table 2 Coefficients estimation results (10158401015840)

1198861 1198871 1198862 1198872

120575120579 minus4209 minus6664 7793 6057120575120574 minus6414 4108 minus0127 minus9708

they can be used to correct and compensate pitch and rolloutput

From Figure 16 it can be seen that the fluctuation of 120575120579120575120574mainly presents second harmonic frequency with respectto rotation period Consequently it can be modeled by (16)and the coefficients estimation results are shown in Table 2Consider

120575120579 = 119886

1205791 sin120593+ 1198871205791 cos120593+ 1198861205792 sin 2120593+ 1198871205792 cos 2120593

120575120574 = 119886

1205741 sin120593+ 1198871205741 cos120593+ 1198861205742 sin 2120593+ 1198871205742 cos 2120593(16)

Figure 17 shows the fitting residuals of 120575120579 120575120574 and theyare less than 510158401015840 Then the transformation matrix 119862119904

119887

can becalculated according to (10) and (11) Thus the pitch and rolloutput compensation results can be acquired as shown inFigure 18

Conclusions can be drawn from Figure 18 that after pitchand roll output compensation the peak-to-peak pitch and rollerrors decreased from 20sim30 arc seconds to less than 5 arcseconds which is improved nearly 5 times It is proved thatthe proposed output compensation algorithm for pitch androll is valid and the pitch and roll short-term output accuracyis improved obviously

However it is worth mentioning that the compensationalgorithm here is not only suitable for single-axis rotationRINS For dual-axis and tri-axis RINS it is inevitable tohave encoder installation eccentricity and other mentionedproblems meanwhile the irregular rotation is hard to avoidas well consequently the analysis and attitude compensationand correction algorithm presented in this paper are alsosuitable for other types of RINS

5 Conclusion

This paper researched the attitude output accuracy improve-ment in rotation RINS Comparative experiment resultsindicate that velocity and position accuracy gains greatimprovement in rotation mode while attitude accuracy iseven worse than that in strapdown mode The reasons thatcause attitude output accuracy loss are analyzed and thena new attitude output compensation algorithm for RINS ispresented The experimental results proved validity of theproposed attitude correction and compensation algorithmwith short-term pitch and roll output accuracy improvedfrom 20sim30 arc seconds to less than 5 arc seconds andazimuth output accuracy improved from 2sim3 arc minutes toless than 05 arc minutes

According to dead reckoning principle high accuracy invelocity and position should be matched with high attitudeaccuracy The proposed compensation algorithm in thispaper solved the problem of attitude accuracy loss of RINSin practical applications which is significant for many task

Mathematical Problems in Engineering 9

0 60 120 180 240 300 360minus20

0

20

40

OriginalFitted

OriginalFitted

0 60 120 180 240 300 360minus20

0

20

40

120575120574

(998400998400)

120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 16 Relationships between 120575120579 120575120574 and rotation angle

0 60 120 180 240 300 360minus20

0

20

40

0 60 120 180 240 300 360minus20

0

20

40

Resid

uals

of120575120574

(998400998400)

Resid

uals

of120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 17 Fitting residuals of 120575120579 120575120574

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

OriginalCompensated

OriginalCompensated

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

Pitc

h er

ror (

998400998400)

Roll

erro

r (998400998400

)

Figure 18 Comparison of original and compensated pitch and roll errors

systems where attitude accuracy is urgently required Inaddition the proposed compensation algorithm is a generalmethod it is not only suitable for single-axis RINS but it canalso be used to improve short-term attitude output accuracyin dual-axis and tri-axis RINS

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported and funded by the National Nat-ural Science Foundation of China (L142200032) and Long-Term Development Strategic Research of China EngineeringScience and Technology (2014-zcq-01) The authors appreci-ate the support and fund

References

[1] E S Geller ldquoInertial system platform rotationrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol AES-4 no 4 pp557ndash568 1968

[2] E Levinson and R Majure ldquoAccuracy enhancement techniquesapplied to the marine ring laser inertial navigator (MARLIN)rdquoNavigation vol 34 no 1 pp 64ndash86 1987

[3] E Levinson J ter Horst and M Willcocks ldquoThe next genera-tion marine inertial navigator is here nowrdquo in Proceedings of thePosition Location and Navigation Symposium pp 121ndash127 IEEEApril 1994

[4] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation Journal ofthe Institute of Navigation vol 28 no 4 pp 311ndash341 1981

[5] Y Yang and L-J Miao ldquoFiber-optic strapdown inertial systemwith sensing cluster continuous rotationrdquo IEEE Transactions onAerospace and Electronic Systems vol 40 no 4 pp 1173ndash11782004

10 Mathematical Problems in Engineering

[6] S Ishibashi S Tsukioka H Yoshida et al ldquoAccuracy improve-ment of an inertial navigation system brought about by therotational motionrdquo in Proceedings of the OCEANSmdashEurope pp1ndash5 IEEE Aberdeen Scotland June 2007

[7] S Ishibashi S Tsukioka T Sawa et al ldquoThe rotation controlsystem to improve the accuracy of an inertial navigation systeminstalled in an autonomous underwater vehiclerdquo in Proceedingsof the Symposium on Underwater Technology and Workshop onScientific Use of Submarine Cables and Related Technologies pp495ndash498 IEEE Tokyo Japan April 2007

[8] A Li G-B Chang F-J Qin and H-W Li ldquoImproved precisionof strapdown inertial navigation system brought by dual-axiscontinuous rotation of inertial measurement unitrdquo in Proceed-ings of the 2nd International Asia Conference on Informatics inControl Automation and Robotics (CAR rsquo10) vol 1 pp 284ndash287March 2010

[9] G Chang J Xu A Li and K Cheng ldquoError analysis andsimulation of the dual-axis rotation-dwell autocompensatingstrapdown inertial navigation systemrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo10) pp 124ndash127 March 2010

[10] Z Deng M Sun and B Wang ldquoError modulation schemeanalysis of dual-axis rotating strap-down inertial navigationsystem based on FOGrdquo in Proceedings of the 33rd ChineseControl Conference (CCC rsquo14) pp 692ndash696Nanjing China July2014

[11] C Jianhua LMingyue CDaidai C Li and S Junyu ldquoResearchof strapdown inertial navigation system monitor techniquebased on dual-axis consequential rotationrdquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 203ndash208 June 2011

[12] R B Morrow Jr and D W Heckman ldquoHigh precision IFOGinsertion into the strategic submarine navigation systemrdquo inProceedings of the IEEE Position Location and Navigation Sym-posium pp 332ndash338 IEEE Palm Springs Calif USA April1998

[13] D W Heckman and L M Baretela ldquoImproved affordabilityof high precision submarine inertial navigation by insertion ofrapidly developing fiber optic gyro technologyrdquo in Proceedingsof the IEEE Position Location and and Navigation Symposiumpp 404ndash410 March 2000

[14] S Qin Z Huang and X Wang ldquoOptical angular encoderinstallation error measurement and calibration by ring lasergyroscoperdquo IEEETransactions on Instrumentation andMeasure-ment vol 59 no 3 pp 506ndash511 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Actual rotation axis

Rotation axis

ys

Ay

Az

xs

xs

AxAx

zszb

ysAz

Ay 120575120579

120575120574

zb(zs)

Figure 9 Comparison of ideal rotation and actual irregular rotation

0 500 1000 1500 2000 2500 3000 3500 4000minus004

minus002

0

002

004

Time (s)0 500 1000 1500 2000 2500 3000 3500 4000

minus004

minus002

0

002

004

Time (s)

Acce

lero

met

er x

(ms2)

Acce

lero

met

er y

(ms2)

Figure 10 Initial outputs of horizontal accelerometers

0 500 1000 1500 2000 2500 3000 3500 4000minus200

minus100

0

100

200

Time (s)0 500 1000 1500 2000 2500 3000 3500 4000

minus200

minus100

0

100

200

Time (s)

Resid

ual o

f acc

elero

met

er120583

g)

Resid

ual o

f acc

eler

omet

er120583

g)

x (

y (

Figure 11 Residual components of horizontal accelerometers

be standard sine or cosine form with the same period asIMU rotation and the amplitude is determined by initialpitch and roll of the rotation plane However due to thefluctuation of rotation plane in actual situation the projectionof acceleration of gravity on horizontal accelerometers willbring about irregular variation and the deviation betweenthe rotation plane and the fixed plane can be representedby irregular variation in horizontal accelerometers Figure 10shows the original output of two horizontal accelerometers inactual experiments In order to obtain the irregular rotationcomponents firstly the projection of acceleration of gravitycaused by pitch and roll should be deducted from thehorizontal accelerometers according to

119886

1015840

119909

= 119886

119909+1198921205740 cos120593minus1198921205790 sin120593

119886

1015840

119910

= 119886

119910minus1198921205790 cos120593minus1198921205740 sin120593

(8)

where 1205790 1205740 are initial pitch and roll acquired by thealignment process in RINS

Figure 11 shows the residual components of horizon-tal accelerometers during the experiment Both horizon-tal accelerometers residual components mainly representsecond harmonic frequency related to the rotation periodand show good repeatability during 4 bidirectional rotationperiods In a single bidirectional period the fluctuationson the horizontal accelerometers are also axial symmetryThe fluctuation amplitude is about 150 120583g which means themaximum deviation between the rotation plane and the fixedplane is about 30 arc seconds and it coincides with theamplitude of pitch and roll output fluctuation in Figure 6

The residuals of horizontal accelerometers can be trans-formed into deviation angles between the rotation plane and

Mathematical Problems in Engineering 7

120575120574

120575120574

120575120579

120575120579O

zs

xs

ys

y998400s

x998400s

z998400s

Figure 12 Transformation relation between s frame and 1199041015840 frame

the fixed plane which are defined as 120575120579 120575120574 as shown inFigure 9 (right) and are given as follows

120575120579 =

119886

1015840

119910

119892

120575120574 = minus

119886

1015840

119909

119892

(9)

To compensate influence caused by 120575120579 120575120574 anothercoordinate frame should be defined in the proposed RINS todistinguish actual irregular rotation from ideal rotation Theideal rotation frame is named as 1199041015840 frame whose O-1199111015840

119904

axis iscoinciding with O-zb axis all the time The previous 119904 frameis the actual rotation frame representing the instantaneousrotation frame and it fluctuates around the ideal fixed planeTaking into consideration irregular rotation attitude matrixobtained by (6) should be described as 119862119899

1199041015840 The transforma-

tion matrix from s frame to 1199041015840 frame can be acquired by 120575120579120575120574 and then the output attitude matrix 119862119899

119887

can be updated by

119862

119899

119887

= 119862

119899

1199041015840119862

119904

1015840

119904

119862

119904

119887

(10)

Figure 12 shows the relationship between s frame and 1199041015840

frameThe transformationmatrix1198621199041015840

119904

can be obtained by twoEuler angle rotations according to

119862

119904

1015840

119904

= 119877

119910(120575120574) 119877

119909(120575120579)

(11)

where 119877

119910(120575120574) = [

cos 120575120574 0 minus sin 1205751205740 1 0

sin 120575120574 0 cos 120575120574] and 119877

119909(120575120579) =

[

1 0 00 cos 120575120579 sin 1205751205790 minus sin 120575120579 cos 120575120579

]Since deviation angles 120575120579 120575120574 between 119904 frame and 119904

1015840

frame are small the matrix 1198621199041015840

119904

can be simplified as follows

119862

119904

1015840

119904

asymp

[

[

[

1 0 minus120575120574

0 1 120575120579

120575120574 minus120575120579 1

]

]

]

(12)

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

OriginalFitted

Angle (∘)

Enco

der a

ngle

erro

r (998400998400

)

Figure 13 Relationship between encoder angle error and rotationangle

4 RINS Attitude Error Compensation andExperimental Verification

According to the previous analysis both encoder angle errorsand axis irregular rotation can be corrected and then RINSattitude error fluctuation can be mostly compensated thusimproving short-term attitude output accuracy

41 AzimuthOutput Correction Experimental Results in RINSIt can be seen in Figure 8 that 120575120593 is axial symmetry aboutforward and reverse rotation thus it is closely related toencoder angle 120593 and the relationship between 120575120593 and 120593 isshown in Figure 13

Figure 13 indicates that the fluctuation of 120575120593 presents fun-damental frequencywith respect to the rotation angle Conse-quently the error variation can be compensated through datafitting by the mathematical model given in

120575120593 = 1198960 + 1198961 sin120593+ 1198962 cos120593 (13)

Use 119883 to denote state variable where 119883 = [1198960 1198961 1198962]119879

and then the coefficients 1198960 1198961 1198962 can be acquired throughleast square fitting algorithm by the following measurementsequation

119885 = 119867119883 (14)

where 119885 = [1205751205931 1205751205932 sdot sdot sdot 120575120593

119898]

119879 119867 = [

1 sin1205931 cos12059311 sin1205932 cos1205932sdotsdotsdot sdotsdotsdot sdotsdotsdot

1 sin120593119898

cos120593119898

]

119898times3

and119898 denotes the total number of useful measurementsSince the coefficients 1198960 1198961 1198962 fitting results are acquired

(in this RINS prototype the coefficients fitting results are1198960 = 0032 1198961 = minus6647 and 1198962 = 1249) the encoder angleerror in RINS could be corrected in real time by correctingcoefficients 1198960 1198961 1198962 according to

8 Mathematical Problems in Engineering

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

Angle (∘)

Resid

uals

(998400998400)

Figure 14 Fitting residuals of encoder angle errors

0 500 1000 1500 2000 2500 3000 3500 4000minus120

minus80

minus40

0

40

80

120

160

Time (s)

OriginalCompensated

Azi

mut

h er

ror (

998400998400)

Figure 15 Comparison of original and compensated azimutherrors

120593

119862= 120593minus (1198960 + 1198961 sin120593+ 1198962 cos120593) (15)

where 120593119862denotes the encoder angle to be compensated

Figure 14 shows the fitting residuals of encoder angleerrors and it is less than 2010158401015840 Then the transformation matrix119862

119904

119887

can be calculated by 120593119862instead of 120593 The azimuth output

correction results and comparisons are shown in Figure 15It can be seen from Figure 15 that after azimuth out-

put correction the peak-to-peak azimuth errors decreasedfrom 2sim3 arc minutes to less than 05 arc minutes whichis improved nearly 5 times and it means the short-termazimuth output accuracy achieves significantly improvement

42 Pitch and Roll Compensation Results in RINS The actualexperimental data also show that being similar to encoderangle errors the derivation angles 120575120579 120575120574 acquired by resid-uals of horizontal accelerometers are axial symmetry withrespect to encoder angle 120593 which are shown in Figure 16 and

Table 2 Coefficients estimation results (10158401015840)

1198861 1198871 1198862 1198872

120575120579 minus4209 minus6664 7793 6057120575120574 minus6414 4108 minus0127 minus9708

they can be used to correct and compensate pitch and rolloutput

From Figure 16 it can be seen that the fluctuation of 120575120579120575120574mainly presents second harmonic frequency with respectto rotation period Consequently it can be modeled by (16)and the coefficients estimation results are shown in Table 2Consider

120575120579 = 119886

1205791 sin120593+ 1198871205791 cos120593+ 1198861205792 sin 2120593+ 1198871205792 cos 2120593

120575120574 = 119886

1205741 sin120593+ 1198871205741 cos120593+ 1198861205742 sin 2120593+ 1198871205742 cos 2120593(16)

Figure 17 shows the fitting residuals of 120575120579 120575120574 and theyare less than 510158401015840 Then the transformation matrix 119862119904

119887

can becalculated according to (10) and (11) Thus the pitch and rolloutput compensation results can be acquired as shown inFigure 18

Conclusions can be drawn from Figure 18 that after pitchand roll output compensation the peak-to-peak pitch and rollerrors decreased from 20sim30 arc seconds to less than 5 arcseconds which is improved nearly 5 times It is proved thatthe proposed output compensation algorithm for pitch androll is valid and the pitch and roll short-term output accuracyis improved obviously

However it is worth mentioning that the compensationalgorithm here is not only suitable for single-axis rotationRINS For dual-axis and tri-axis RINS it is inevitable tohave encoder installation eccentricity and other mentionedproblems meanwhile the irregular rotation is hard to avoidas well consequently the analysis and attitude compensationand correction algorithm presented in this paper are alsosuitable for other types of RINS

5 Conclusion

This paper researched the attitude output accuracy improve-ment in rotation RINS Comparative experiment resultsindicate that velocity and position accuracy gains greatimprovement in rotation mode while attitude accuracy iseven worse than that in strapdown mode The reasons thatcause attitude output accuracy loss are analyzed and thena new attitude output compensation algorithm for RINS ispresented The experimental results proved validity of theproposed attitude correction and compensation algorithmwith short-term pitch and roll output accuracy improvedfrom 20sim30 arc seconds to less than 5 arc seconds andazimuth output accuracy improved from 2sim3 arc minutes toless than 05 arc minutes

According to dead reckoning principle high accuracy invelocity and position should be matched with high attitudeaccuracy The proposed compensation algorithm in thispaper solved the problem of attitude accuracy loss of RINSin practical applications which is significant for many task

Mathematical Problems in Engineering 9

0 60 120 180 240 300 360minus20

0

20

40

OriginalFitted

OriginalFitted

0 60 120 180 240 300 360minus20

0

20

40

120575120574

(998400998400)

120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 16 Relationships between 120575120579 120575120574 and rotation angle

0 60 120 180 240 300 360minus20

0

20

40

0 60 120 180 240 300 360minus20

0

20

40

Resid

uals

of120575120574

(998400998400)

Resid

uals

of120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 17 Fitting residuals of 120575120579 120575120574

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

OriginalCompensated

OriginalCompensated

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

Pitc

h er

ror (

998400998400)

Roll

erro

r (998400998400

)

Figure 18 Comparison of original and compensated pitch and roll errors

systems where attitude accuracy is urgently required Inaddition the proposed compensation algorithm is a generalmethod it is not only suitable for single-axis RINS but it canalso be used to improve short-term attitude output accuracyin dual-axis and tri-axis RINS

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported and funded by the National Nat-ural Science Foundation of China (L142200032) and Long-Term Development Strategic Research of China EngineeringScience and Technology (2014-zcq-01) The authors appreci-ate the support and fund

References

[1] E S Geller ldquoInertial system platform rotationrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol AES-4 no 4 pp557ndash568 1968

[2] E Levinson and R Majure ldquoAccuracy enhancement techniquesapplied to the marine ring laser inertial navigator (MARLIN)rdquoNavigation vol 34 no 1 pp 64ndash86 1987

[3] E Levinson J ter Horst and M Willcocks ldquoThe next genera-tion marine inertial navigator is here nowrdquo in Proceedings of thePosition Location and Navigation Symposium pp 121ndash127 IEEEApril 1994

[4] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation Journal ofthe Institute of Navigation vol 28 no 4 pp 311ndash341 1981

[5] Y Yang and L-J Miao ldquoFiber-optic strapdown inertial systemwith sensing cluster continuous rotationrdquo IEEE Transactions onAerospace and Electronic Systems vol 40 no 4 pp 1173ndash11782004

10 Mathematical Problems in Engineering

[6] S Ishibashi S Tsukioka H Yoshida et al ldquoAccuracy improve-ment of an inertial navigation system brought about by therotational motionrdquo in Proceedings of the OCEANSmdashEurope pp1ndash5 IEEE Aberdeen Scotland June 2007

[7] S Ishibashi S Tsukioka T Sawa et al ldquoThe rotation controlsystem to improve the accuracy of an inertial navigation systeminstalled in an autonomous underwater vehiclerdquo in Proceedingsof the Symposium on Underwater Technology and Workshop onScientific Use of Submarine Cables and Related Technologies pp495ndash498 IEEE Tokyo Japan April 2007

[8] A Li G-B Chang F-J Qin and H-W Li ldquoImproved precisionof strapdown inertial navigation system brought by dual-axiscontinuous rotation of inertial measurement unitrdquo in Proceed-ings of the 2nd International Asia Conference on Informatics inControl Automation and Robotics (CAR rsquo10) vol 1 pp 284ndash287March 2010

[9] G Chang J Xu A Li and K Cheng ldquoError analysis andsimulation of the dual-axis rotation-dwell autocompensatingstrapdown inertial navigation systemrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo10) pp 124ndash127 March 2010

[10] Z Deng M Sun and B Wang ldquoError modulation schemeanalysis of dual-axis rotating strap-down inertial navigationsystem based on FOGrdquo in Proceedings of the 33rd ChineseControl Conference (CCC rsquo14) pp 692ndash696Nanjing China July2014

[11] C Jianhua LMingyue CDaidai C Li and S Junyu ldquoResearchof strapdown inertial navigation system monitor techniquebased on dual-axis consequential rotationrdquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 203ndash208 June 2011

[12] R B Morrow Jr and D W Heckman ldquoHigh precision IFOGinsertion into the strategic submarine navigation systemrdquo inProceedings of the IEEE Position Location and Navigation Sym-posium pp 332ndash338 IEEE Palm Springs Calif USA April1998

[13] D W Heckman and L M Baretela ldquoImproved affordabilityof high precision submarine inertial navigation by insertion ofrapidly developing fiber optic gyro technologyrdquo in Proceedingsof the IEEE Position Location and and Navigation Symposiumpp 404ndash410 March 2000

[14] S Qin Z Huang and X Wang ldquoOptical angular encoderinstallation error measurement and calibration by ring lasergyroscoperdquo IEEETransactions on Instrumentation andMeasure-ment vol 59 no 3 pp 506ndash511 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

120575120574

120575120574

120575120579

120575120579O

zs

xs

ys

y998400s

x998400s

z998400s

Figure 12 Transformation relation between s frame and 1199041015840 frame

the fixed plane which are defined as 120575120579 120575120574 as shown inFigure 9 (right) and are given as follows

120575120579 =

119886

1015840

119910

119892

120575120574 = minus

119886

1015840

119909

119892

(9)

To compensate influence caused by 120575120579 120575120574 anothercoordinate frame should be defined in the proposed RINS todistinguish actual irregular rotation from ideal rotation Theideal rotation frame is named as 1199041015840 frame whose O-1199111015840

119904

axis iscoinciding with O-zb axis all the time The previous 119904 frameis the actual rotation frame representing the instantaneousrotation frame and it fluctuates around the ideal fixed planeTaking into consideration irregular rotation attitude matrixobtained by (6) should be described as 119862119899

1199041015840 The transforma-

tion matrix from s frame to 1199041015840 frame can be acquired by 120575120579120575120574 and then the output attitude matrix 119862119899

119887

can be updated by

119862

119899

119887

= 119862

119899

1199041015840119862

119904

1015840

119904

119862

119904

119887

(10)

Figure 12 shows the relationship between s frame and 1199041015840

frameThe transformationmatrix1198621199041015840

119904

can be obtained by twoEuler angle rotations according to

119862

119904

1015840

119904

= 119877

119910(120575120574) 119877

119909(120575120579)

(11)

where 119877

119910(120575120574) = [

cos 120575120574 0 minus sin 1205751205740 1 0

sin 120575120574 0 cos 120575120574] and 119877

119909(120575120579) =

[

1 0 00 cos 120575120579 sin 1205751205790 minus sin 120575120579 cos 120575120579

]Since deviation angles 120575120579 120575120574 between 119904 frame and 119904

1015840

frame are small the matrix 1198621199041015840

119904

can be simplified as follows

119862

119904

1015840

119904

asymp

[

[

[

1 0 minus120575120574

0 1 120575120579

120575120574 minus120575120579 1

]

]

]

(12)

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

OriginalFitted

Angle (∘)

Enco

der a

ngle

erro

r (998400998400

)

Figure 13 Relationship between encoder angle error and rotationangle

4 RINS Attitude Error Compensation andExperimental Verification

According to the previous analysis both encoder angle errorsand axis irregular rotation can be corrected and then RINSattitude error fluctuation can be mostly compensated thusimproving short-term attitude output accuracy

41 AzimuthOutput Correction Experimental Results in RINSIt can be seen in Figure 8 that 120575120593 is axial symmetry aboutforward and reverse rotation thus it is closely related toencoder angle 120593 and the relationship between 120575120593 and 120593 isshown in Figure 13

Figure 13 indicates that the fluctuation of 120575120593 presents fun-damental frequencywith respect to the rotation angle Conse-quently the error variation can be compensated through datafitting by the mathematical model given in

120575120593 = 1198960 + 1198961 sin120593+ 1198962 cos120593 (13)

Use 119883 to denote state variable where 119883 = [1198960 1198961 1198962]119879

and then the coefficients 1198960 1198961 1198962 can be acquired throughleast square fitting algorithm by the following measurementsequation

119885 = 119867119883 (14)

where 119885 = [1205751205931 1205751205932 sdot sdot sdot 120575120593

119898]

119879 119867 = [

1 sin1205931 cos12059311 sin1205932 cos1205932sdotsdotsdot sdotsdotsdot sdotsdotsdot

1 sin120593119898

cos120593119898

]

119898times3

and119898 denotes the total number of useful measurementsSince the coefficients 1198960 1198961 1198962 fitting results are acquired

(in this RINS prototype the coefficients fitting results are1198960 = 0032 1198961 = minus6647 and 1198962 = 1249) the encoder angleerror in RINS could be corrected in real time by correctingcoefficients 1198960 1198961 1198962 according to

8 Mathematical Problems in Engineering

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

Angle (∘)

Resid

uals

(998400998400)

Figure 14 Fitting residuals of encoder angle errors

0 500 1000 1500 2000 2500 3000 3500 4000minus120

minus80

minus40

0

40

80

120

160

Time (s)

OriginalCompensated

Azi

mut

h er

ror (

998400998400)

Figure 15 Comparison of original and compensated azimutherrors

120593

119862= 120593minus (1198960 + 1198961 sin120593+ 1198962 cos120593) (15)

where 120593119862denotes the encoder angle to be compensated

Figure 14 shows the fitting residuals of encoder angleerrors and it is less than 2010158401015840 Then the transformation matrix119862

119904

119887

can be calculated by 120593119862instead of 120593 The azimuth output

correction results and comparisons are shown in Figure 15It can be seen from Figure 15 that after azimuth out-

put correction the peak-to-peak azimuth errors decreasedfrom 2sim3 arc minutes to less than 05 arc minutes whichis improved nearly 5 times and it means the short-termazimuth output accuracy achieves significantly improvement

42 Pitch and Roll Compensation Results in RINS The actualexperimental data also show that being similar to encoderangle errors the derivation angles 120575120579 120575120574 acquired by resid-uals of horizontal accelerometers are axial symmetry withrespect to encoder angle 120593 which are shown in Figure 16 and

Table 2 Coefficients estimation results (10158401015840)

1198861 1198871 1198862 1198872

120575120579 minus4209 minus6664 7793 6057120575120574 minus6414 4108 minus0127 minus9708

they can be used to correct and compensate pitch and rolloutput

From Figure 16 it can be seen that the fluctuation of 120575120579120575120574mainly presents second harmonic frequency with respectto rotation period Consequently it can be modeled by (16)and the coefficients estimation results are shown in Table 2Consider

120575120579 = 119886

1205791 sin120593+ 1198871205791 cos120593+ 1198861205792 sin 2120593+ 1198871205792 cos 2120593

120575120574 = 119886

1205741 sin120593+ 1198871205741 cos120593+ 1198861205742 sin 2120593+ 1198871205742 cos 2120593(16)

Figure 17 shows the fitting residuals of 120575120579 120575120574 and theyare less than 510158401015840 Then the transformation matrix 119862119904

119887

can becalculated according to (10) and (11) Thus the pitch and rolloutput compensation results can be acquired as shown inFigure 18

Conclusions can be drawn from Figure 18 that after pitchand roll output compensation the peak-to-peak pitch and rollerrors decreased from 20sim30 arc seconds to less than 5 arcseconds which is improved nearly 5 times It is proved thatthe proposed output compensation algorithm for pitch androll is valid and the pitch and roll short-term output accuracyis improved obviously

However it is worth mentioning that the compensationalgorithm here is not only suitable for single-axis rotationRINS For dual-axis and tri-axis RINS it is inevitable tohave encoder installation eccentricity and other mentionedproblems meanwhile the irregular rotation is hard to avoidas well consequently the analysis and attitude compensationand correction algorithm presented in this paper are alsosuitable for other types of RINS

5 Conclusion

This paper researched the attitude output accuracy improve-ment in rotation RINS Comparative experiment resultsindicate that velocity and position accuracy gains greatimprovement in rotation mode while attitude accuracy iseven worse than that in strapdown mode The reasons thatcause attitude output accuracy loss are analyzed and thena new attitude output compensation algorithm for RINS ispresented The experimental results proved validity of theproposed attitude correction and compensation algorithmwith short-term pitch and roll output accuracy improvedfrom 20sim30 arc seconds to less than 5 arc seconds andazimuth output accuracy improved from 2sim3 arc minutes toless than 05 arc minutes

According to dead reckoning principle high accuracy invelocity and position should be matched with high attitudeaccuracy The proposed compensation algorithm in thispaper solved the problem of attitude accuracy loss of RINSin practical applications which is significant for many task

Mathematical Problems in Engineering 9

0 60 120 180 240 300 360minus20

0

20

40

OriginalFitted

OriginalFitted

0 60 120 180 240 300 360minus20

0

20

40

120575120574

(998400998400)

120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 16 Relationships between 120575120579 120575120574 and rotation angle

0 60 120 180 240 300 360minus20

0

20

40

0 60 120 180 240 300 360minus20

0

20

40

Resid

uals

of120575120574

(998400998400)

Resid

uals

of120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 17 Fitting residuals of 120575120579 120575120574

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

OriginalCompensated

OriginalCompensated

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

Pitc

h er

ror (

998400998400)

Roll

erro

r (998400998400

)

Figure 18 Comparison of original and compensated pitch and roll errors

systems where attitude accuracy is urgently required Inaddition the proposed compensation algorithm is a generalmethod it is not only suitable for single-axis RINS but it canalso be used to improve short-term attitude output accuracyin dual-axis and tri-axis RINS

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported and funded by the National Nat-ural Science Foundation of China (L142200032) and Long-Term Development Strategic Research of China EngineeringScience and Technology (2014-zcq-01) The authors appreci-ate the support and fund

References

[1] E S Geller ldquoInertial system platform rotationrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol AES-4 no 4 pp557ndash568 1968

[2] E Levinson and R Majure ldquoAccuracy enhancement techniquesapplied to the marine ring laser inertial navigator (MARLIN)rdquoNavigation vol 34 no 1 pp 64ndash86 1987

[3] E Levinson J ter Horst and M Willcocks ldquoThe next genera-tion marine inertial navigator is here nowrdquo in Proceedings of thePosition Location and Navigation Symposium pp 121ndash127 IEEEApril 1994

[4] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation Journal ofthe Institute of Navigation vol 28 no 4 pp 311ndash341 1981

[5] Y Yang and L-J Miao ldquoFiber-optic strapdown inertial systemwith sensing cluster continuous rotationrdquo IEEE Transactions onAerospace and Electronic Systems vol 40 no 4 pp 1173ndash11782004

10 Mathematical Problems in Engineering

[6] S Ishibashi S Tsukioka H Yoshida et al ldquoAccuracy improve-ment of an inertial navigation system brought about by therotational motionrdquo in Proceedings of the OCEANSmdashEurope pp1ndash5 IEEE Aberdeen Scotland June 2007

[7] S Ishibashi S Tsukioka T Sawa et al ldquoThe rotation controlsystem to improve the accuracy of an inertial navigation systeminstalled in an autonomous underwater vehiclerdquo in Proceedingsof the Symposium on Underwater Technology and Workshop onScientific Use of Submarine Cables and Related Technologies pp495ndash498 IEEE Tokyo Japan April 2007

[8] A Li G-B Chang F-J Qin and H-W Li ldquoImproved precisionof strapdown inertial navigation system brought by dual-axiscontinuous rotation of inertial measurement unitrdquo in Proceed-ings of the 2nd International Asia Conference on Informatics inControl Automation and Robotics (CAR rsquo10) vol 1 pp 284ndash287March 2010

[9] G Chang J Xu A Li and K Cheng ldquoError analysis andsimulation of the dual-axis rotation-dwell autocompensatingstrapdown inertial navigation systemrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo10) pp 124ndash127 March 2010

[10] Z Deng M Sun and B Wang ldquoError modulation schemeanalysis of dual-axis rotating strap-down inertial navigationsystem based on FOGrdquo in Proceedings of the 33rd ChineseControl Conference (CCC rsquo14) pp 692ndash696Nanjing China July2014

[11] C Jianhua LMingyue CDaidai C Li and S Junyu ldquoResearchof strapdown inertial navigation system monitor techniquebased on dual-axis consequential rotationrdquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 203ndash208 June 2011

[12] R B Morrow Jr and D W Heckman ldquoHigh precision IFOGinsertion into the strategic submarine navigation systemrdquo inProceedings of the IEEE Position Location and Navigation Sym-posium pp 332ndash338 IEEE Palm Springs Calif USA April1998

[13] D W Heckman and L M Baretela ldquoImproved affordabilityof high precision submarine inertial navigation by insertion ofrapidly developing fiber optic gyro technologyrdquo in Proceedingsof the IEEE Position Location and and Navigation Symposiumpp 404ndash410 March 2000

[14] S Qin Z Huang and X Wang ldquoOptical angular encoderinstallation error measurement and calibration by ring lasergyroscoperdquo IEEETransactions on Instrumentation andMeasure-ment vol 59 no 3 pp 506ndash511 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0 60 120 180 240 300 360minus80

minus60

minus40

minus20

0

20

40

60

80

Angle (∘)

Resid

uals

(998400998400)

Figure 14 Fitting residuals of encoder angle errors

0 500 1000 1500 2000 2500 3000 3500 4000minus120

minus80

minus40

0

40

80

120

160

Time (s)

OriginalCompensated

Azi

mut

h er

ror (

998400998400)

Figure 15 Comparison of original and compensated azimutherrors

120593

119862= 120593minus (1198960 + 1198961 sin120593+ 1198962 cos120593) (15)

where 120593119862denotes the encoder angle to be compensated

Figure 14 shows the fitting residuals of encoder angleerrors and it is less than 2010158401015840 Then the transformation matrix119862

119904

119887

can be calculated by 120593119862instead of 120593 The azimuth output

correction results and comparisons are shown in Figure 15It can be seen from Figure 15 that after azimuth out-

put correction the peak-to-peak azimuth errors decreasedfrom 2sim3 arc minutes to less than 05 arc minutes whichis improved nearly 5 times and it means the short-termazimuth output accuracy achieves significantly improvement

42 Pitch and Roll Compensation Results in RINS The actualexperimental data also show that being similar to encoderangle errors the derivation angles 120575120579 120575120574 acquired by resid-uals of horizontal accelerometers are axial symmetry withrespect to encoder angle 120593 which are shown in Figure 16 and

Table 2 Coefficients estimation results (10158401015840)

1198861 1198871 1198862 1198872

120575120579 minus4209 minus6664 7793 6057120575120574 minus6414 4108 minus0127 minus9708

they can be used to correct and compensate pitch and rolloutput

From Figure 16 it can be seen that the fluctuation of 120575120579120575120574mainly presents second harmonic frequency with respectto rotation period Consequently it can be modeled by (16)and the coefficients estimation results are shown in Table 2Consider

120575120579 = 119886

1205791 sin120593+ 1198871205791 cos120593+ 1198861205792 sin 2120593+ 1198871205792 cos 2120593

120575120574 = 119886

1205741 sin120593+ 1198871205741 cos120593+ 1198861205742 sin 2120593+ 1198871205742 cos 2120593(16)

Figure 17 shows the fitting residuals of 120575120579 120575120574 and theyare less than 510158401015840 Then the transformation matrix 119862119904

119887

can becalculated according to (10) and (11) Thus the pitch and rolloutput compensation results can be acquired as shown inFigure 18

Conclusions can be drawn from Figure 18 that after pitchand roll output compensation the peak-to-peak pitch and rollerrors decreased from 20sim30 arc seconds to less than 5 arcseconds which is improved nearly 5 times It is proved thatthe proposed output compensation algorithm for pitch androll is valid and the pitch and roll short-term output accuracyis improved obviously

However it is worth mentioning that the compensationalgorithm here is not only suitable for single-axis rotationRINS For dual-axis and tri-axis RINS it is inevitable tohave encoder installation eccentricity and other mentionedproblems meanwhile the irregular rotation is hard to avoidas well consequently the analysis and attitude compensationand correction algorithm presented in this paper are alsosuitable for other types of RINS

5 Conclusion

This paper researched the attitude output accuracy improve-ment in rotation RINS Comparative experiment resultsindicate that velocity and position accuracy gains greatimprovement in rotation mode while attitude accuracy iseven worse than that in strapdown mode The reasons thatcause attitude output accuracy loss are analyzed and thena new attitude output compensation algorithm for RINS ispresented The experimental results proved validity of theproposed attitude correction and compensation algorithmwith short-term pitch and roll output accuracy improvedfrom 20sim30 arc seconds to less than 5 arc seconds andazimuth output accuracy improved from 2sim3 arc minutes toless than 05 arc minutes

According to dead reckoning principle high accuracy invelocity and position should be matched with high attitudeaccuracy The proposed compensation algorithm in thispaper solved the problem of attitude accuracy loss of RINSin practical applications which is significant for many task

Mathematical Problems in Engineering 9

0 60 120 180 240 300 360minus20

0

20

40

OriginalFitted

OriginalFitted

0 60 120 180 240 300 360minus20

0

20

40

120575120574

(998400998400)

120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 16 Relationships between 120575120579 120575120574 and rotation angle

0 60 120 180 240 300 360minus20

0

20

40

0 60 120 180 240 300 360minus20

0

20

40

Resid

uals

of120575120574

(998400998400)

Resid

uals

of120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 17 Fitting residuals of 120575120579 120575120574

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

OriginalCompensated

OriginalCompensated

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

Pitc

h er

ror (

998400998400)

Roll

erro

r (998400998400

)

Figure 18 Comparison of original and compensated pitch and roll errors

systems where attitude accuracy is urgently required Inaddition the proposed compensation algorithm is a generalmethod it is not only suitable for single-axis RINS but it canalso be used to improve short-term attitude output accuracyin dual-axis and tri-axis RINS

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported and funded by the National Nat-ural Science Foundation of China (L142200032) and Long-Term Development Strategic Research of China EngineeringScience and Technology (2014-zcq-01) The authors appreci-ate the support and fund

References

[1] E S Geller ldquoInertial system platform rotationrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol AES-4 no 4 pp557ndash568 1968

[2] E Levinson and R Majure ldquoAccuracy enhancement techniquesapplied to the marine ring laser inertial navigator (MARLIN)rdquoNavigation vol 34 no 1 pp 64ndash86 1987

[3] E Levinson J ter Horst and M Willcocks ldquoThe next genera-tion marine inertial navigator is here nowrdquo in Proceedings of thePosition Location and Navigation Symposium pp 121ndash127 IEEEApril 1994

[4] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation Journal ofthe Institute of Navigation vol 28 no 4 pp 311ndash341 1981

[5] Y Yang and L-J Miao ldquoFiber-optic strapdown inertial systemwith sensing cluster continuous rotationrdquo IEEE Transactions onAerospace and Electronic Systems vol 40 no 4 pp 1173ndash11782004

10 Mathematical Problems in Engineering

[6] S Ishibashi S Tsukioka H Yoshida et al ldquoAccuracy improve-ment of an inertial navigation system brought about by therotational motionrdquo in Proceedings of the OCEANSmdashEurope pp1ndash5 IEEE Aberdeen Scotland June 2007

[7] S Ishibashi S Tsukioka T Sawa et al ldquoThe rotation controlsystem to improve the accuracy of an inertial navigation systeminstalled in an autonomous underwater vehiclerdquo in Proceedingsof the Symposium on Underwater Technology and Workshop onScientific Use of Submarine Cables and Related Technologies pp495ndash498 IEEE Tokyo Japan April 2007

[8] A Li G-B Chang F-J Qin and H-W Li ldquoImproved precisionof strapdown inertial navigation system brought by dual-axiscontinuous rotation of inertial measurement unitrdquo in Proceed-ings of the 2nd International Asia Conference on Informatics inControl Automation and Robotics (CAR rsquo10) vol 1 pp 284ndash287March 2010

[9] G Chang J Xu A Li and K Cheng ldquoError analysis andsimulation of the dual-axis rotation-dwell autocompensatingstrapdown inertial navigation systemrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo10) pp 124ndash127 March 2010

[10] Z Deng M Sun and B Wang ldquoError modulation schemeanalysis of dual-axis rotating strap-down inertial navigationsystem based on FOGrdquo in Proceedings of the 33rd ChineseControl Conference (CCC rsquo14) pp 692ndash696Nanjing China July2014

[11] C Jianhua LMingyue CDaidai C Li and S Junyu ldquoResearchof strapdown inertial navigation system monitor techniquebased on dual-axis consequential rotationrdquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 203ndash208 June 2011

[12] R B Morrow Jr and D W Heckman ldquoHigh precision IFOGinsertion into the strategic submarine navigation systemrdquo inProceedings of the IEEE Position Location and Navigation Sym-posium pp 332ndash338 IEEE Palm Springs Calif USA April1998

[13] D W Heckman and L M Baretela ldquoImproved affordabilityof high precision submarine inertial navigation by insertion ofrapidly developing fiber optic gyro technologyrdquo in Proceedingsof the IEEE Position Location and and Navigation Symposiumpp 404ndash410 March 2000

[14] S Qin Z Huang and X Wang ldquoOptical angular encoderinstallation error measurement and calibration by ring lasergyroscoperdquo IEEETransactions on Instrumentation andMeasure-ment vol 59 no 3 pp 506ndash511 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 60 120 180 240 300 360minus20

0

20

40

OriginalFitted

OriginalFitted

0 60 120 180 240 300 360minus20

0

20

40

120575120574

(998400998400)

120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 16 Relationships between 120575120579 120575120574 and rotation angle

0 60 120 180 240 300 360minus20

0

20

40

0 60 120 180 240 300 360minus20

0

20

40

Resid

uals

of120575120574

(998400998400)

Resid

uals

of120575120579

(998400998400)

Angle (∘) Angle (∘)

Figure 17 Fitting residuals of 120575120579 120575120574

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

OriginalCompensated

OriginalCompensated

0 500 1000 1500 2000 2500 3000 3500 4000minus40

minus20

0

20

40

Time (s)

Pitc

h er

ror (

998400998400)

Roll

erro

r (998400998400

)

Figure 18 Comparison of original and compensated pitch and roll errors

systems where attitude accuracy is urgently required Inaddition the proposed compensation algorithm is a generalmethod it is not only suitable for single-axis RINS but it canalso be used to improve short-term attitude output accuracyin dual-axis and tri-axis RINS

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported and funded by the National Nat-ural Science Foundation of China (L142200032) and Long-Term Development Strategic Research of China EngineeringScience and Technology (2014-zcq-01) The authors appreci-ate the support and fund

References

[1] E S Geller ldquoInertial system platform rotationrdquo IEEE Transac-tions on Aerospace and Electronic Systems vol AES-4 no 4 pp557ndash568 1968

[2] E Levinson and R Majure ldquoAccuracy enhancement techniquesapplied to the marine ring laser inertial navigator (MARLIN)rdquoNavigation vol 34 no 1 pp 64ndash86 1987

[3] E Levinson J ter Horst and M Willcocks ldquoThe next genera-tion marine inertial navigator is here nowrdquo in Proceedings of thePosition Location and Navigation Symposium pp 121ndash127 IEEEApril 1994

[4] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation Journal ofthe Institute of Navigation vol 28 no 4 pp 311ndash341 1981

[5] Y Yang and L-J Miao ldquoFiber-optic strapdown inertial systemwith sensing cluster continuous rotationrdquo IEEE Transactions onAerospace and Electronic Systems vol 40 no 4 pp 1173ndash11782004

10 Mathematical Problems in Engineering

[6] S Ishibashi S Tsukioka H Yoshida et al ldquoAccuracy improve-ment of an inertial navigation system brought about by therotational motionrdquo in Proceedings of the OCEANSmdashEurope pp1ndash5 IEEE Aberdeen Scotland June 2007

[7] S Ishibashi S Tsukioka T Sawa et al ldquoThe rotation controlsystem to improve the accuracy of an inertial navigation systeminstalled in an autonomous underwater vehiclerdquo in Proceedingsof the Symposium on Underwater Technology and Workshop onScientific Use of Submarine Cables and Related Technologies pp495ndash498 IEEE Tokyo Japan April 2007

[8] A Li G-B Chang F-J Qin and H-W Li ldquoImproved precisionof strapdown inertial navigation system brought by dual-axiscontinuous rotation of inertial measurement unitrdquo in Proceed-ings of the 2nd International Asia Conference on Informatics inControl Automation and Robotics (CAR rsquo10) vol 1 pp 284ndash287March 2010

[9] G Chang J Xu A Li and K Cheng ldquoError analysis andsimulation of the dual-axis rotation-dwell autocompensatingstrapdown inertial navigation systemrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo10) pp 124ndash127 March 2010

[10] Z Deng M Sun and B Wang ldquoError modulation schemeanalysis of dual-axis rotating strap-down inertial navigationsystem based on FOGrdquo in Proceedings of the 33rd ChineseControl Conference (CCC rsquo14) pp 692ndash696Nanjing China July2014

[11] C Jianhua LMingyue CDaidai C Li and S Junyu ldquoResearchof strapdown inertial navigation system monitor techniquebased on dual-axis consequential rotationrdquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 203ndash208 June 2011

[12] R B Morrow Jr and D W Heckman ldquoHigh precision IFOGinsertion into the strategic submarine navigation systemrdquo inProceedings of the IEEE Position Location and Navigation Sym-posium pp 332ndash338 IEEE Palm Springs Calif USA April1998

[13] D W Heckman and L M Baretela ldquoImproved affordabilityof high precision submarine inertial navigation by insertion ofrapidly developing fiber optic gyro technologyrdquo in Proceedingsof the IEEE Position Location and and Navigation Symposiumpp 404ndash410 March 2000

[14] S Qin Z Huang and X Wang ldquoOptical angular encoderinstallation error measurement and calibration by ring lasergyroscoperdquo IEEETransactions on Instrumentation andMeasure-ment vol 59 no 3 pp 506ndash511 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[6] S Ishibashi S Tsukioka H Yoshida et al ldquoAccuracy improve-ment of an inertial navigation system brought about by therotational motionrdquo in Proceedings of the OCEANSmdashEurope pp1ndash5 IEEE Aberdeen Scotland June 2007

[7] S Ishibashi S Tsukioka T Sawa et al ldquoThe rotation controlsystem to improve the accuracy of an inertial navigation systeminstalled in an autonomous underwater vehiclerdquo in Proceedingsof the Symposium on Underwater Technology and Workshop onScientific Use of Submarine Cables and Related Technologies pp495ndash498 IEEE Tokyo Japan April 2007

[8] A Li G-B Chang F-J Qin and H-W Li ldquoImproved precisionof strapdown inertial navigation system brought by dual-axiscontinuous rotation of inertial measurement unitrdquo in Proceed-ings of the 2nd International Asia Conference on Informatics inControl Automation and Robotics (CAR rsquo10) vol 1 pp 284ndash287March 2010

[9] G Chang J Xu A Li and K Cheng ldquoError analysis andsimulation of the dual-axis rotation-dwell autocompensatingstrapdown inertial navigation systemrdquo in Proceedings of theInternational Conference on Measuring Technology and Mecha-tronics Automation (ICMTMA rsquo10) pp 124ndash127 March 2010

[10] Z Deng M Sun and B Wang ldquoError modulation schemeanalysis of dual-axis rotating strap-down inertial navigationsystem based on FOGrdquo in Proceedings of the 33rd ChineseControl Conference (CCC rsquo14) pp 692ndash696Nanjing China July2014

[11] C Jianhua LMingyue CDaidai C Li and S Junyu ldquoResearchof strapdown inertial navigation system monitor techniquebased on dual-axis consequential rotationrdquo in Proceedings of theInternational Conference on Information and Automation (ICIArsquo11) pp 203ndash208 June 2011

[12] R B Morrow Jr and D W Heckman ldquoHigh precision IFOGinsertion into the strategic submarine navigation systemrdquo inProceedings of the IEEE Position Location and Navigation Sym-posium pp 332ndash338 IEEE Palm Springs Calif USA April1998

[13] D W Heckman and L M Baretela ldquoImproved affordabilityof high precision submarine inertial navigation by insertion ofrapidly developing fiber optic gyro technologyrdquo in Proceedingsof the IEEE Position Location and and Navigation Symposiumpp 404ndash410 March 2000

[14] S Qin Z Huang and X Wang ldquoOptical angular encoderinstallation error measurement and calibration by ring lasergyroscoperdquo IEEETransactions on Instrumentation andMeasure-ment vol 59 no 3 pp 506ndash511 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of