ErrorMechanismandSelf-CalibrationofSingle-AxisRotational...

Research ArticleError Mechanism and Self-Calibration of Single-Axis RotationalInertial Navigation System

Bo Nie Guiming Chen Xianting Luo and Bingqi Liu

High-Tech Institute of Xirsquoan Xirsquoan 710025 China

Correspondence should be addressed to Bingqi Liu iqgnibuil126com

Received 19 January 2019 Revised 28 May 2019 Accepted 6 September 2019 Published 30 September 2019

Academic Editor Luis M Lopez-Ochoa

Copyright copy 2019 Bo Nie et al is 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

With the rapid development of inertial technology the rotational inertial system has been widely used for high-accuracynavigation In addition the single-axis rotational inertial navigation system is one of the most popular navigation systems dueto its low cost However the fact which cannot be ignored is that although single axis rotational inertial navigation system canrestrain divergence of attitude error it cannot eliminate velocity error which is mainly caused by the gyro misalignment andscale factor error related to rotational axis A novel calibration method established on filter has been proposed Positionvelocity and attitude are chosen to be the state variables In order to estimate the gyro errors related to rotational axis theseerrors must be included in the state equation Correspondingly zero velocity and rate of turntable are chosen to be measuredSimulation has been carried out to verify the correctness of the theory and the real test has been performed to furtherdemonstrate the validity of the method By compensating the main error estimated by filter it can be found that the ac-cumulated errors in velocity and attitude are decreased So the precision of navigation is greatly improved with theproposed method

1 Introduction

Inertial navigation system (INS) can obtain position atti-tude and velocity of the craft by resolving the data sampledby its inertial measurement unit which is IMU and containsthree orthogonal gyros and accelerometers [1]us the INSis self-contained and widely used in military field such asairplanes submarines and ships INS can be divided intotwo categories platform and strap-down inertial navigationsystem Compared with platform INS strap-down INS hasthe characteristics of higher reliability smaller volumebetter maintenance and lower cost [2ndash4]

Device error is affected by the precision of the naviga-tion Meanwhile velocity and attitude errors will accumulatewith time quickly [5] ere are two major approaches toenhance the precision of inertial navigation One is to im-prove the performance of gyro and accelerometer Howeverthis method is expensive and restricted by technique eother one is error compensation Compensation can beperformed based on the external information such as Global

Positioning System (GPS) and turntable [6] Self-compen-sation is another way to eliminate the device error Rotationmodulation technology has been one of themost widely usedself-compensation methods especially in the field of aero-space for its low cost and reliability

In rotational inertial navigation system (RINS) IMU isplaced on the indexing mechanization which is fixed on thecraft [7ndash9] us the self-compensation can be performedby designing appropriate rotating sequence RINS can bedivided into single-axis dual-axis and triaxial-axis or above[10ndash12]

e advantage of rotation modulation is that it canrealize self-compensation of sensor errors during navigationprocess [13 14] However errors still exist and affect theprecision of the navigation

In triaxis RINS instrument errors such as accelerometernonlinearity and inner lever-arm can be accurately com-pensated through optimal estimation with velocity andposition error measurements Meanwhile proper rotationscheme design can help modulate and estimate error

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 8912341 11 pageshttpsdoiorg10115520198912341

parameters of IMU Taking commercial cost into consid-eration triaxis and dual-axis RINSs are more expensive thansingle-axis [15ndash18] In consequence single-axis RINS will bemainly discussed in this paper

In single-axis RINS gyro drift is the main concernShang et al [19] pointed out that single-axis rotation canmodulate the constant drift perpendicular to the rotationalaxis while gyro drift on the spin axis still diverge with timeIn order to deal with the constant drift of spin axis iden-tification methods of gyro drift have been proposed Yu et al[20] calibrated the single-axis gyro drift by using radial basisfunction network A least square method was established tocalibrate the axial gyro drift precisely by Hu et al [21]Similar methods of calibration of gyro drifts are shown in[22ndash24]

Besides the gyro drift error propagation of the single axisrotational INS showed that scale factor error and mis-alignment errors related to the rotational axis cannot bemodulated [25 26] e scale factor error along the rota-tional axis was the main error that affected the single-axisrotational INS in [27] However no paper can give answer tothis question erefore a novel method which can calibratethe instrument errors is proposed in this article

Compared with other methods the method proposed inthis paper is suitable for single-axis RINS to calibrate maindevice errors by using zero velocity and attitude errorsuscompensation can be made to reduce the error accumulationduring the navigation process Based on this the precision ofthe single-axis RINS can be promoted

e advantages of the proposed method are shown asfollows according to single-axis rotational inertial navi-gation system this paper provides the deduction of errorpropagation in terms of misalignment and scale factorerror related to rotational axis Besides it points outthat in single-axis RINS attitude error caused by gyromisalignment related to rotational axis can be modulatedbut velocity error still exists Simulation has been carriedout to verify this thesis By observing the velocity andrate of turntable gyro misalignment and scale factorerrors of the rotational axis can be estimated under sta-tionary base

e paper is organized as follows Section 2 provides theerror mechanism of single axis rotational INS Section 3introduces the design of filter for single-axis rotationalsystem Section 4 describes the process of the simulationwhich is carried out to verify the error mechanism and theproposed method In order to further demonstrate theproposed method experiment has been performed andresults are shown in Section 5 Conclusion is drawn inSection 6

2 Error Mechanism

e denotation of frames is shown in Figure 1 where i

denotes earth-centered inertial frame e denotes earth-centered frame n denotes navigation frame and b denotesbody frame

21 Wander Azimuth Frame e wander frame is widelyused in navigation system and it can be obtained by rotatingwith respect to the local geodetic frame z axis by the wanderframe angle α

In this paper considering the fact calibration is per-formed on stationary base α is chosen to be zero e errormodel in local geodetic frame can be considered the same aswander azimuth frame in a short time

22 Error Analysis To further demonstrate the errormechanism in this paper simplification and assumption aregiven

e IMU can be rotated about the vertical body axis withconstant angular velocity ω

δωbib 1113957ωb

ib minus ωbib

gBx

gBy

gBz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

Sx Mxy Mxz

Myx Sy Myz

Mzx Mzy Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ωbibx

ωbiby

ωbibz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)

e gyro angular velocity error equation is shownabove where 1113957ωb

ib denotes the measured angular velocity inthe body frame ωb

ib denotes the true angular velocity δωbib

denotes the angular velocity error gBi denotes gyro biasSi denotes the scale factor error and Mij denotes themisalignment

ωbeb 0 0 ω1113858 1113859

T

ωbib ωb

ie + ωbeb

(2)

where ωbeb denotes the angular velocity provided by turn-

table which rotates around the body spin axis with constantrate ω and ωb

ie denotes the earth angular velocity in the bodyframe

Gyro error model is given by equation (1) Consideringthe angular velocity provided by turntable is much largerthan the earth angular velocity and gyro bias the approx-imation of gyro model is shown as follows

Xn

Yn

Zn

αα

XY

Z

Figure 1 Wander azimuth frame

2 Mathematical Problems in Engineering

δωbib asymp

Sx Mxy Mxz

Myx Sy Myz

Mzx Mzy Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ωbiex

ωbiey

ωbiez + ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

asymp

Sx Mxy Mxz

Myx Sy Myz

Mzx Mzy Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ω middot

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)

e simplified gyro error model shows that the mostimportant error sources are Mxz Myz Sz Correspondinglythe velocity error accumulated by inertial device error willdirectly affect the precision of navigation e rotation se-quence from navigation frame to body frame is yaw pitchand roll respectively

Cnb C

ns middot C

sb (4)

Cns

cos c0 cosφG0 minus sin c0 sin θ0 sinφG0 minus cos θ0 sinφG0 sin c0 cosφG0 + cos c0 sin θ0 sinφG0

cos c0 sinφG0 + sin c0 sin θ0 cosφG0 cos θ0 cosφG0 sin c0 sinφG0 minus cos c0 sin θ0 cosφG0

minus sin c0 cos θ0 sin θ0 cos c0 cos θ0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Csb

cos ζ minus sin ζ 0sin ζ cos ζ 00 0 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)

where Csb is the direction cosine matrix of body frame Cn

s isused to transfer the initial body frame to the wander azimuthnavigation frame φG0 denotes the initial yaw angle c0denotes the initial roll angle θ0 denotes the initial pitchangle and ζ denotes rotation angle around spin axis frominitial frame s to body frame b

e attitude error can be defined as (concrete deductionis shown in Appendix)

ϕ(t) ϕ(0) + δϕ(t)

ϕ(0) + Cns

Mxz sinωt + Myz cosωt minus Myz

minus Mxz cosωt + Myz sinωt + Mxz

Szωt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)

where ϕ(0) denotes the attitude error at the beginning andδϕ(t) denotes the accumulated attitude error From equa-tion (6) it can be seen that the accumulated leveling attitudeerror can be modulated by 2π rotation However the initialattitude error ϕ(0) still exists and has great effect on thevelocity error

Besides scale factor error of the rotational axis gyro willaffect yaw angle e velocity error dynamic equationsdemonstrate the relationship between attitude error andvelocity error as follows

δ _vx δfx minus gϕy

δ _vy δfy + gϕx(7)

Velocity errors can be obtained by the integral ofequation (7) To research the factors that affect velocityexpansion of integral is shown as follows (concrete de-duction is given in Appendix)

δvx(t) δvx(0) + 1113946t

0δfx minus gϕy1113872 1113873dt

1113874δvx(0) minus Cns (2 1)

g

ω1113874 minus Mxz cosωt

+ Myz sinωt + Mxz1113875 + Cns (2 2)

g

ω1113874Mxz sinωt

+ Myz cosωt minus Myz11138751113875 + 1113874 minus gϕy(0)t

+ Cns (2 1)gMyzt minus C

ns (2 2)gMxzt

minus Cns (2 3)

g

2Szωt

21113875

(8)

From the final expansion of equation (8) it can be seenthat terms in the first bracket are converged because trig-onometric function is bounded Terms in the second bracketare diverging with time us the drift of velocity δvx(t) ismainly caused by the items in the second bracket thatcontains initial attitude error gyro misalignment and scalefactor error related to rotational axis

Similarly the error velocity δvy(t) can be expressed as

Mathematical Problems in Engineering 3

δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

δvy(0) + Cns (1 1)

g

ωminus Mxz cosωt + Myz sinωt + Mxz1113872 1113873 minus C

ns (1 2)

g

ωMxz sinωt + Myz cosωt minus Myz1113872 11138731113874 1113875

+ gϕx(0)t minus Cns (1 1)gMyzt + C

ns (1 2)gMxzt + C

ns (1 3)

g

2Szωt

21113874 1113875

(9)

Obviously the main angular velocity error above whichis induced by scale factor error and misalignment errorrelated to body spin axis cannot be modulated during therotation e navigation attitude error will accumulatequickly because of this us the precision of the INSnavigation system can be improved by using the self-cali-bration method to estimate the errors related to body spinaxis

3 Design of Filter for Single-AxisRotational System

According to the error mechanism the proposed methodshould estimate the instrument errors including scale factorerrors and misalignment of the related rotation axis Besidesthe navigation information position velocity and attitudeshould also be contained in the measurement matrixCorrespondingly the measurement model of the proposedmethod should be established by parameters that can beobtained precisely

31 State Equation e state model chosen in this paper islisted as follows

X(t) δθx δθy δh δvx δvy δvz ψx ψy ψz Mxz Myz Sz1113960 1113961T

(10)

where δθx δθy and δh denote position errors δvx δvy andδvz denote velocity errors ψx ψy and ψz denote attitudeerrors and Mxz Myz and Sz denote deterministic unknownerrors

(1) Instrument error equationse gyro errors are modelled as constant during thefiltering process

_Mxz 0

_Myz 0

_Sz 0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(11)

(2) Attitude error equations_ψx ωzψy minus ωyψz minus εx

prime

_ψy minus ωzψx + ωxψz minus εyprime

_ψz ωyψx minus ωxψy minus εzprime

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

ω ρ +Ω

(12)

where _ψi denotes dynamic error εiprime denotes gyro

error under navigation system ρ denotes the navi-gation to earth frame angular rotation rate in thenavigation frame andΩ denotes the Earthrsquos rotationrate

(3) Velocity error equations

δ _vx minus gδθy + 2Ωzδvy minus (ρ + 2Ω)yvz minus fzψy + fyψz + nablax

δ _vy gδθx minus 2Ωzδvx minus (ρ + 2Ω)xvz + fzψx minus fxψz + nablay

δ _vz 2g

Rδh + ωy +Ωy1113872 1113873δvx minus ωx +Ωx( 1113857δvy minus fyψx

+fxψy + nablaz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)

where fi denotes the sensed output of i-axis accel-erometer δθx and δθy denote the angular positionerrors δh denotes the altitude error and nablai denotesthe error of accelerometer

(4) Position error equations

δ _θx minusvz

Rδθx minus

1Rδvy

δ _θy minusvz

Rδθy +

1Rδvx

δ _h δvz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

where R denotes the earth radius(5) Gyro error model

εx

εy

εz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

n

Cnb

Bx

By

Bz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

Sx Mxy Mxz

Myx Sy Myz

Mzx Mzy Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦middot

ωxib

ωy

ib

ωzib

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(15)

where εi denotes the angular velocity error caused bygyro Bi denotes the bias of i-axis gyro Si denotes thescale factor error of i-axis gyro and Mij denotes themisalignment error of gyro in i-axis related to j-axis

32 Measurement Model e angular velocity provided byrate turntable is much larger than that of the earth so the


errors are mainly caused by scale factor errors and mis-alignment errors related to rotational axis us the error ofgyro output and the velocity during navigation process canbe utilized as observations to estimate the correspondingerrors

(1) Given that the IMU is tested on stationary base thevelocity of IMU should be zero e output velocityof IMU can be defined as the measurement Z1

Z1 δV Vins minus 0

δvx

δvy

δvz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)

e measurement equation of sensed velocity of testedIMU can be described as

Z1(t) H1X(t) + η (17)

where the measurement matrix is

H1

01times3 1 0 0 01times6



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (18)

(2) e attitude error always exists and the computedattitude matrix translating from navigation frame tobody frame is represented as

1113957Cb

n (I minus ϕtimes)Cbn (19)

where ϕ denotes tilt error and Cbn denotes attitude

direction cosine matrix without attitude error einput rate of turntable and the rate of earth rotation canbe considered as references and the output rate of IMUcan be measured by gyros us the measurement Z2is defined to be the difference between the measuredangular velocity and the true angular velocity

Z2 δωbib 1113957ωb

ib minus 1113957Cb

nωnie minus 1113957ωb

eb

δ1113957ωbib + C

bn ωn

ie times( 1113857ϕn(20)

and themeasurementmatrix ofH2 can be expressed as

H2 03times6 Cbn times ωn

ie

ωbibz 0 0

0 ωbibz 0

0 0 ωbibz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21)

(3) Flow chart of self-calibration filtering is given inFigure 2e principle of the proposed calibration method inthis paper to estimate the unknown errors is shownin Figure 2 Figure 2 uses angular velocity and zerovelocity as the observed quantity A 12-state Kalmanfilter is established by the INS dynamic equationsand simplified gyro error model

4 Simulation

To support the theory discussed above and evaluate theperformance of the proposed method simulations arecarried out in Section 42

41 Condition e IMU errors in this simulation are de-fined as follows gyro bias is 01degh with a white noise of 001deghradicHz accelerometers bias is 100 μg with a white noise of10 μgradicHz misalignment of accelerometers is 20Prime scalefactor error of accelerometer is 100 ppm the misalignmentof gyro about rotational axis (z-axis of body frame) is 60Primeand the others are 20Prime and the scale factor error of gyroaround rotation axis is 300 ppm and the others are 100 ppme main error parameters are listed in Table 1 Datasampling frequency is 50Hz

In order to validate the theory above the initial attitudeangles of roll pitch and yaw are 20deg 30deg and 40deg re-spectively e tested IMU is under stationary base etested IMU rotates with the angular velocity of 10degs and thelength of the test period t is 36 s

42 Simulation Results Assume that there is no attitudeerror at the beginning of navigation and the whole

IMU(gyros accelero

meters)INS solution Kalman filter

Zero velocity

Errors estimates of IMU

Estimated errors

Angular velocity error

Mxz Myz Sz

Mxz Myz Sz

Figure 2 Flow chart of self-calibration filter in the proposed method


navigation process lasts 360 s including 10 whole periods tBased on the error sources listed in Tables 1 and 2 thesimulation results are evaluated by 30 Monte Carlos runse state variables of instrument errors are Mxz Myz andSz and the values are 5996Prime5996Prime and 300 ppm re-spectively In Table 3 it is obvious that the estimated

parameters obtained by 12-state filter are very close to thereal values

Figure 3 describes the error covariance root-mean-squarecurve of the state variables during the filtering processe curveshows that the algorithm has better convergence speed and thestate variables Mxz Myz and Sz have good observability

Table 2 Main error sources of gyros

Bias (degh) Misalignment (sec) Scale factor error(ppm) Noise (deghradicHz)

εx εy εz Mxy Mxz Myx Myz Mzx Mzy Sx Sy Sz εn

005 005 005 20 60 20 60 20 20 100 100 300 002

Table 3 Calibration result

Parameter Mxz Myz Sz

Estimate 5996 5996 30002

Table 1 Main error sources of accelerators

Bias (μg) Misalignment (sec) Scale factor error (ppm) Noise (μgradicHz)nablax nablay nablaz Exy Exz Eyx Eyz Ezx Ezy Kx Ky Kz nablan

100 100 100 20 20 20 20 20 20 100 100 100 30

0 50 100 150 200 250 300 350 4000

50

100

150

Time (s)

S z (p

pm)

(a)

0 50 100 150 200 250 300 350 4000

5

10

15

20

Time (s)

Mxz

(arc

-sec

)

(b)

0 50 100 150 200 250 300 350 4000

5

10

15

20

Time (s)

Myz

(arc

-sec

)

(c)

Figure 3 Error covariance root-mean-square curve of estimated device error


Figure 4 describes the estimated gyro scale factor errorsand misalignment errors around rotational axis in 30 MonteCarlos runs Similar to error covariance root-mean-squarecurve error estimated curve also converges quickly afterfiltering e steady values of estimated gyro scale factorerrors and misalignment errors can be compensated duringthe navigation process Navigation results after compensa-tion are listed in Table 4

Figure 5 describes the velocity error during navigationprocess without filter e curve shows that velocity erroraccumulates with time quickly which is the same as theunknown attitude error and the inertial device error relatedto rotational axis us the error mechanism deduction hasbeen verified

Table 4 provides a comparison of the navigation resultsunder different conditions rough 360 s navigation withoutcompensation the velocity errorsVx andVy are 2982ms andminus 17548ms and the attitude errors of roll pitch and yawangles are 0467deg 0303deg and 0911deg Meanwhile Table 4provides navigation results under the same condition aftercompensation e velocity errors Vx and Vy are 0065 andminus 0024 and the attitude errors of roll pitch and yaw areminus 0044deg 0034deg and minus 0092deg respectively

us simulation results show that velocity errorcaused by scale factor errors and misalignment errorsrelated to rotation axis cannot be modulated by the

single-axis rotational inertial navigation system With theproposed method the corresponding errors can becalibrated precisely Simulation results show that navi-gation performance is greatly enhanced by calibrationand compensation

5 Real Test

51 Experiment Set up To further validate the proposedmethod the real calibration test is established e test IMUconsists of three orthogonal gyros whose accuracy is 01deghand accelerometers whose accuracy is 100 μg e test iscarried out with personal computer made by DELL in-dustrial personal computer the self-designed computermonitoring software and navigation computing software

Data are sampled by the frequency of 100Hze turntableshown in Figure 6 is utilized to simulate single-axis rotationalmechanization whose rotation rate accuracy is 00001degs Be-sides its positional accuracy is 2 arc-second Initial angles ofpitch roll and yaw are 00456deg 00279deg and 00197deg e ro-tation rate of turntable is 10degs corresponding to the simulation

In the beginning the turntable remains stationary for140 s to collect data for initial alignment en the IMUrotates at the angular velocity of 10degs around its z-axis atleast 900 s for navigation e turntable should be rotated toits original position after navigation

0 50 100 150 200 250 300 350 4000

100

200

300

400

Time (s)

S z (p

pm)

(a)

0 50 100 150 200 250 300 350 4000

20

40

60

80

Time (s)

Mxz

(arc

-sec

)

(b)

0 50 100 150 200 250 300 350 4000

20

40

60

80

Time (s)

Myz

(arc

-sec

)

(c)

Figure 4 Estimation curve of device error related to rotational axis


52 Experiment Result Gyro errors of rotation axis areestimated by the designed self-calibration filter Estimatedvalues of Mxz Myz and Sz are shown in Figure 3 It isobvious that estimation curve converges very quickly esteady values of estimated scale factor errors and mis-alignment errors Mxz Myz and Sz related to rotational axisare utilized as calibration results Calibration results esti-mated by filter are listed in Table 5 where Mxz is 1542PrimeMyz is minus 1077Prime and Sz is 840 ppm

Figure 6 shows the navigation results of accumulatedvelocity errors before compensation It is evident that ve-locity accumulates quickly and it can be proved that single-axis rotational INS cannot modulate scale factor error andmisalignment errors related to rotation axis Figure 7 dis-plays the velocity estimated curve after compensating for thecorresponding errors Compared with the estimated curvebefore compensation the error accumulates more slowly It

proves that the proposed method can decrease or eliminatethe effects caused by gyro in rotation axis Figure 8 shows theVx Vy curve after compensation

Navigation results about velocity and attitude are listed inTable 6 Before compensation the velocity errors in level driftare 49222ms and minus 19773ms respectively while aftercompensation are 02950ms and 05888ms In terms ofattitude errors the improvement after compensation is alsoobvious Compared with the initial attitude the attitude er-rors accumulated during navigation and resolved by single-axis rotational INS in pitch roll and yaw are 00426 degree01037 degree and 08712 degree Similarly attitude errors inpitch roll and yaw angle after compensation are00011 degree 00834 degree and 00771 degree Data expressthat the precision of navigation is enhanced at least 20

us the proposed method with filter can estimate thedevice error related to rotational axis effectively e

0 200 400 600 800 1000 1200ndash2

0

2

4

6

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash3

ndash2

ndash1

0

1

Time (s)

V y (m

s)

(b)

Figure 6 Velocity error of pure navigation before compensation



Estimated 1542511 minus 1077430 840126

Table 4 Pure navigation results

Parameter Before compensation After compensation eory valueVx (ms) 2982 00447 0Vy (ms) 17548 01184 0Vz (mx) 0 0 0Pitch (deg) 20467 19954 20Roll (deg) 29697 30030 30Yaw (deg) 40911 39910 40

0 50 100 150 200 250 300 350 400

1

0

2

3

Time (s)

V x (m

s)

(a)

0 50 100 150 200 250 300 350 400ndash10

0

10

20

Time (s)

V y (m

s)

(b)

Figure 5 Velocity error curve resolved by single axis rotational INS without compensation


precision of the single axis rotational INS is improved bycompensating the estimated steady value of correspondingerrors

6 Conclusions

In the single-axis rotational inertial navigation system in-strument errors related to rotational axis cannot be mod-ulated and these errors will have great impact on thenavigation precision Error mechanism and equations ofvelocity and attitude have been shown in this paper and acalibration method with filter has been proposed e statemodel should contain device errors which cannot bemodulated by rotation In the stationary base zero velocityand the known turntable angular velocity can be chosen asthe measurement IMU errors can be estimated and com-pensated online to reduce the navigation error Simulationhas been carried out to verify the correctness of errormechanism and the feasibility of the proposed method the

0 200 400 600 800 1000 1200ndash02

0

02

04

06

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash1

ndash05

0

05

Time (s)

V y (m

s)

(b)

Figure 8 Velocity errors of pure navigation after compensation

Table 6 Pure navigation results in experiment

Parameter eoryvalue

Beforecompensation

Aftercompensation

Vx (ms) 0 49222 02950Vy (ms) 0 minus 19773 05888Vz (mx) 0 0 0Pitch(deg) 00456 00043 00467

Roll (deg) 00279 minus 00758 minus 00455Yaw (deg) 00197 08809 00968

0 200 400 600 800 1000 12000

50

100

Time (s)

S z (p

pm)

(a)

0 200 400 600 800 1000 12000

100

200

Time (s)

Mxz

(arc

-sec

)

(b)

0 200 400 600 800 1000 1200ndash200

ndash100

0

Time (s)

Myz

(arc

-sec

)

(c)

Figure 7 Errors estimation curve related to rotational axis gyro


experiment results further illustrate the validation of theproposed method and the precision of the single-axis ro-tational INS can be improved

It is shown that the precision of navigation is enhancedat least 20 e proposed method using filter can estimatethe device error related to rotational axis effectively eprecision of the single axis rotational INS has been improvedby compensating the estimated steady value of corre-sponding errors

Appendix

Deduction of equation (7)

ϕ(t) ϕ(0) + δϕ(t) (A1)

In the equation δϕ(t) can be approximated as

δϕ(t) 1113946t

0C

nbδω

bibdt

asymp 1113946t

0C

nb

minus minus Mxz

minus minus Myz

minus minus Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dt 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

(A2)

where φ ωt then inserting equation (5) into ϕ(t) we get

ϕ(t) ϕ(0) + δϕ(t)

ϕ(0) + 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns 1113946

ωt

o

Mxz cosφ minus Myz sinφ

Mxz sinφ + Myz cosφ

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns



Szωt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A3)

Deduction of equation (8) the velocity error accumu-lated with time can be written as the integral of equation (7)

δvx(t) δvx(0) + 1113946t


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

(A4)

Normally accelerometer error δf can be modulated by2π rotation under stationary base us the simplified ve-locity model can be approximately written as

δvx(t) δvx(0) minus g 1113946t

0ϕydt

δvy(t) δvy(0) + g 1113946t

0ϕxdt

(A5)

Combining with equation (7) equation (A5) can beexpressed as


0ϕydt

δvx(0) minus g 1113946t

01113874ϕy(0) + C

ns (2 1)

middot Mxz sinωt + Myz cosωt minus Myz1113872 1113873

+ Cns (2 2) minus Mxz cosωt + Myz sinωt + Mxz1113872 1113873

+ Cns (2 3)Szωt1113875dt

δvx(0) minus gϕy(0)t minus Cns (2 1)g

middotminus Mxz cosωt

ω+

Mxz

ω+

Myz sinωt

ωminus Myzt1113888 1113889

minus Cns (2 2)g1113888

minus Mxz sinωt

ω+

Myz

ωminus

Myz cosωt

ω

+ Mxzt1113889 minus Cns (2 3)

g

2Szωt

2

1113874δvx(0) minus Cns (2 1)

g

ω1113874 minus Mxz cosωt + Myz sinωt

+ Mxz1113875 + Cns (2 2)

g

ω1113874Mxz sinωt + Myz cosωt

minus Myz11138751113875 + 1113874 minus gϕy(0)t + Cns (2 1)gMyzt

minus Cns (2 2)gMxzt minus C

ns (2 3)

g

2Szωt

21113875

(A6)


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

1113874δvy(0) + Cns (1 1)

g


+ Mxz1113875 minus Cns (1 2)

g


minus Myz11138751113875 + 1113874gϕx(0)t minus Cns (1 1)gMyzt

+ Cns (1 2)gMxzt + C

ns (1 3)

g

2Szωt

21113875

(A7)

Data Availability

Due to project team constraints other data materials cannotbe provided for the time being


Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] J D Watson and F H C Crick ldquoMolecular structure ofnucleic acids a structure for deoxyribose nucleic acidrdquo Na-ture vol 171 no 4356 pp 737-738 1953

[2] E Foxlin ldquoPedestrian tracking with shoe-mounted inertialsensorsrdquo IEEE Computer Graphics and Applications vol 25no 6 pp 38ndash46 2005

[3] J Prieto S Mazuelas and M Z Win ldquoContext-aided inertialnavigation via belief condensationrdquo IEEE Transactions onSignal Processing vol 64 no 12 pp 3250ndash3261 2016

[4] D H Titterton and J L Weston ldquoStrapdown inertial navi-gation technologyrdquo Aerospace amp Electronic Systems MagazineIEEE vol 20 no 7 pp 33-34 2004

[5] S Mazuelas Y Yuan Shen and M Z Win ldquoBelief con-densation filteringrdquo IEEE Transactions on Signal Processingvol 61 no 18 pp 4403ndash4415 2013

[6] Y Bar-Shalom T Kirubarajan and X R Li Estimation withApplications to Tracking and Navigation Deory Algorithmsand Software John Wiley amp Sons Hoboken NJ USA 2001

[7] E S Geller ldquoInertial system platform rotationrdquo IEEETransactions on Aerospace and Electronic Systems vol AES-4no 4 pp 557ndash568 1968

[8] P Lv J Lai J Liu and M Nie ldquoe compensation effects ofgyrosrsquo stochastic errors in a rotational inertial navigationsystemrdquo Journal of Navigation vol 67 no 6 pp 1069ndash10882014

[9] W Gao Y Zhang and J Wang ldquoResearch on initial align-ment and self-calibration of rotary strapdown inertial navi-gation systemsrdquo Sensors vol 15 no 2 pp 3154ndash3171 2015

[10] Q Ren B Wang Z Deng and M Fu ldquoA multi-position self-calibration method for dual-axis rotational inertial navigationsystemrdquo Sensors and Actuators A Physical vol 219 pp 24ndash312014

[11] F Yu and S S Yuan ldquoDual-axis rotating scheme for fiberstrap-down inertial navigation systemrdquo Journal of HarbinEngineering University vol 35 no 12 pp 1ndash7 2014

[12] F Sun and Y Q Wang ldquoRotation scheme design for mod-ulated SINS with three rotating axesrdquo Chinese Journal ofScientific Instrument vol 34 no 1 pp 65ndash72 2013

[13] B Yuan D Liao and S Han ldquoError compensation of anoptical gyro INS by multi-axis rotationrdquoMeasurement Scienceand Technology vol 23 no 2 Article ID 025102 2012

[14] Q Zhang L Wang Z Liu and P Feng ldquoAn accurate cali-bration method based on velocity in a rotational inertialnavigation systemrdquo Sensors vol 15 no 8 pp 18443ndash184582015

[15] P Gao K Li L Wang and Z Liu ldquoA self-calibration methodfor accelerometer nonlinearity errors in triaxis rotationalinertial navigation systemrdquo IEEE Transactions on In-strumentation and Measurement vol 66 no 2 pp 243ndash2532017

[16] T Song K Li and J Sui ldquoZengjun Liu and Juncheng LiuldquoSelf-calibration method of inner lever-arm parameters fortri-axis RINSrdquo Measurement Science and Technology vol 28no 11 Article ID 115105 2017

[17] P Gao K Li L Wang and Z Liu ldquoA self-calibration methodfor tri-axis rotational inertial navigation systemrdquo Measure-ment Science and Technology vol 27 no 11 Article ID115009 2016

[18] P Gao K Li L Wang and J Gao ldquoA self-calibration methodfor non-orthogonal angles of gimbals in tri-axis rotationalinertial navigation systemrdquo IEEE Sensors Journal vol 16no 24 pp 8998ndash9005 2016

[19] S Shang Z Deng M Fu R Wang and F Mo ldquoAnalysis ofsingle-axial rotation modulation influence on FOG INSprecisionrdquo in Proceedings of the 29th Chinese Control Con-ference IEEE Beijing China July 2010

[20] X D Yu X Wei Y Li and X Long ldquoApplication of radialbasis function network for identification of axial RLG drifts insingle-axis rotation inertial navigation systemrdquo Journal ofNational University of Defense Technology vol 34 no 3pp 48ndash52 2012

[21] J Hu X H Cheng and Y X Zhu ldquoCalibration method foraxial gyro drifts of single-axis rotary SINSrdquo Systems Engi-neering and Electronic vol 39 no 7 pp 1564ndash1569 2017

[22] W Lei ldquoIdentification of axial RLG drift in single-axis inertialnavigation system based on artificial fish swarm algorithmrdquoJournal of Applied Optics vol 35 no 4 pp 725ndash728 2014

[23] L C Wang K Li L Wang and J X Gao ldquoIdentifying Z-axisgyro drift and scale factor error using azimuth measurementin fiber optic gyroscope single-axis rotation inertial navigationsystemrdquo Optical Engineering vol 56 no 2 Article ID 0241022017

[24] F Sun J Z Xia B Y Yang et al ldquoFour-position driftmeasurement of SINS based on single-axis rotationrdquo inProceedings of the 2012 IEEEION Position Location andNavigation Symposium (PLANS) IEEE Myrtle Beach SCUSA April 2012

[25] H L Yin G Yang N Song and L Wang ldquoError propagatingcharacteristic analyzing for rotating LG INSrdquo Journal ofBeijing University of Aeronautics and Astronautics vol 38no 3 pp 345ndash350 2012

[26] B L Yuan G Y Rao and D Liao ldquoScale factor error analysisfor rotating inertial navigation systemrdquo Journal of ChineseInertial Technology vol 18 no 2 pp 160ndash164 2010

[27] F Sun Y L Chai Y Y Ben and W Gao ldquoAzimuth rotationexperiment and analysis for fiber-optic gyroscope strapdownsystemrdquo in Proceedings of the 2009 International Conferenceon Information Technology and Computer Science pp 141ndash144 IEEE Kiev Ukraine July 2009


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parameters of IMU Taking commercial cost into consid-eration triaxis and dual-axis RINSs are more expensive thansingle-axis [15ndash18] In consequence single-axis RINS will bemainly discussed in this paper

In single-axis RINS gyro drift is the main concernShang et al [19] pointed out that single-axis rotation canmodulate the constant drift perpendicular to the rotationalaxis while gyro drift on the spin axis still diverge with timeIn order to deal with the constant drift of spin axis iden-tification methods of gyro drift have been proposed Yu et al[20] calibrated the single-axis gyro drift by using radial basisfunction network A least square method was established tocalibrate the axial gyro drift precisely by Hu et al [21]Similar methods of calibration of gyro drifts are shown in[22ndash24]

Besides the gyro drift error propagation of the single axisrotational INS showed that scale factor error and mis-alignment errors related to the rotational axis cannot bemodulated [25 26] e scale factor error along the rota-tional axis was the main error that affected the single-axisrotational INS in [27] However no paper can give answer tothis question erefore a novel method which can calibratethe instrument errors is proposed in this article

Compared with other methods the method proposed inthis paper is suitable for single-axis RINS to calibrate maindevice errors by using zero velocity and attitude errorsuscompensation can be made to reduce the error accumulationduring the navigation process Based on this the precision ofthe single-axis RINS can be promoted

e advantages of the proposed method are shown asfollows according to single-axis rotational inertial navi-gation system this paper provides the deduction of errorpropagation in terms of misalignment and scale factorerror related to rotational axis Besides it points outthat in single-axis RINS attitude error caused by gyromisalignment related to rotational axis can be modulatedbut velocity error still exists Simulation has been carriedout to verify this thesis By observing the velocity andrate of turntable gyro misalignment and scale factorerrors of the rotational axis can be estimated under sta-tionary base

e paper is organized as follows Section 2 provides theerror mechanism of single axis rotational INS Section 3introduces the design of filter for single-axis rotationalsystem Section 4 describes the process of the simulationwhich is carried out to verify the error mechanism and theproposed method In order to further demonstrate theproposed method experiment has been performed andresults are shown in Section 5 Conclusion is drawn inSection 6

2 Error Mechanism

e denotation of frames is shown in Figure 1 where i

denotes earth-centered inertial frame e denotes earth-centered frame n denotes navigation frame and b denotesbody frame

21 Wander Azimuth Frame e wander frame is widelyused in navigation system and it can be obtained by rotatingwith respect to the local geodetic frame z axis by the wanderframe angle α

In this paper considering the fact calibration is per-formed on stationary base α is chosen to be zero e errormodel in local geodetic frame can be considered the same aswander azimuth frame in a short time

22 Error Analysis To further demonstrate the errormechanism in this paper simplification and assumption aregiven

e IMU can be rotated about the vertical body axis withconstant angular velocity ω

δωbib 1113957ωb

ib minus ωbib

gBx

gBy

gBz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

Sx Mxy Mxz

Myx Sy Myz

Mzx Mzy Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ωbibx

ωbiby

ωbibz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)

e gyro angular velocity error equation is shownabove where 1113957ωb

ib denotes the measured angular velocity inthe body frame ωb

ib denotes the true angular velocity δωbib

denotes the angular velocity error gBi denotes gyro biasSi denotes the scale factor error and Mij denotes themisalignment

ωbeb 0 0 ω1113858 1113859

T

ωbib ωb

ie + ωbeb

(2)

where ωbeb denotes the angular velocity provided by turn-

table which rotates around the body spin axis with constantrate ω and ωb

ie denotes the earth angular velocity in the bodyframe

Gyro error model is given by equation (1) Consideringthe angular velocity provided by turntable is much largerthan the earth angular velocity and gyro bias the approx-imation of gyro model is shown as follows

Xn

Yn

Zn

αα

XY

Z

Figure 1 Wander azimuth frame


δωbib asymp

Sx Mxy Mxz

Myx Sy Myz

Mzx Mzy Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ωbiex

ωbiey

ωbiez + ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

asymp

Sx Mxy Mxz

Myx Sy Myz

Mzx Mzy Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ω middot

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)


Cnb C

ns middot C

sb (4)

Cns




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Csb


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)




ϕ(t) ϕ(0) + δϕ(t)

ϕ(0) + Cns



Szωt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)






δvx(t) δvx(0) + 1113946t


1113874δvx(0) minus Cns (2 1)

g


+ Myz sinωt + Mxz1113875 + Cns (2 2)

g

ω1113874Mxz sinωt



ns (2 2)gMxzt

minus Cns (2 3)

g

2Szωt

21113875

(8)




δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

δvy(0) + Cns (1 1)

g


ns (1 2)

g



ns (1 2)gMxzt + C

ns (1 3)

g

2Szωt

21113874 1113875

(9)






(10)



_Mxz 0

_Myz 0

_Sz 0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(11)


prime



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

ω ρ +Ω

(12)






δ _vz 2g


+fxψy + nablaz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)



δ _θx minusvz

Rδθx minus

1Rδvy

δ _θy minusvz

Rδθy +

1Rδvx

δ _h δvz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)


εx

εy

εz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

n

Cnb

Bx

By

Bz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

Sx Mxy Mxz

Myx Sy Myz

Mzx Mzy Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦middot

ωxib

ωy

ib

ωzib

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(15)






Z1 δV Vins minus 0

δvx

δvy

δvz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)


Z1(t) H1X(t) + η (17)


H1




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (18)


1113957Cb





ib minus 1113957Cb


eb

δ1113957ωbib + C

bn ωn

ie times( 1113857ϕn(20)



ie

ωbibz 0 0

0 ωbibz 0

0 0 ωbibz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21)


4 Simulation





IMU(gyros accelero


Zero velocity


Estimated errors


Mxz Myz Sz

Mxz Myz Sz









005 005 005 20 60 20 60 20 20 100 100 300 002



Estimate 5996 5996 30002



100 100 100 20 20 20 20 20 20 100 100 100 30

0 50 100 150 200 250 300 350 4000

50

100

150

Time (s)

S z (p

pm)

(a)

0 50 100 150 200 250 300 350 4000

5

10

15

20

Time (s)

Mxz

(arc

-sec

)

(b)

0 50 100 150 200 250 300 350 4000

5

10

15

20

Time (s)

Myz

(arc

-sec

)

(c)








5 Real Test




0 50 100 150 200 250 300 350 4000

100

200

300

400

Time (s)

S z (p

pm)

(a)

0 50 100 150 200 250 300 350 4000

20

40

60

80

Time (s)

Mxz

(arc

-sec

)

(b)

0 50 100 150 200 250 300 350 4000

20

40

60

80

Time (s)

Myz

(arc

-sec

)

(c)








0 200 400 600 800 1000 1200ndash2

0

2

4

6

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash3

ndash2

ndash1

0

1

Time (s)

V y (m

s)

(b)




Estimated 1542511 minus 1077430 840126



0 50 100 150 200 250 300 350 400

1

0

2

3

Time (s)

V x (m

s)

(a)

0 50 100 150 200 250 300 350 400ndash10

0

10

20

Time (s)

V y (m

s)

(b)




6 Conclusions


0 200 400 600 800 1000 1200ndash02

0

02

04

06

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash1

ndash05

0

05

Time (s)

V y (m

s)

(b)



Parameter eoryvalue

Beforecompensation

Aftercompensation



0 200 400 600 800 1000 12000

50

100

Time (s)

S z (p

pm)

(a)

0 200 400 600 800 1000 12000

100

200

Time (s)

Mxz

(arc

-sec

)

(b)

0 200 400 600 800 1000 1200ndash200

ndash100

0

Time (s)

Myz

(arc

-sec

)

(c)





Appendix


ϕ(t) ϕ(0) + δϕ(t) (A1)


δϕ(t) 1113946t

0C

nbδω

bibdt

asymp 1113946t

0C

nb

minus minus Mxz

minus minus Myz

minus minus Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dt 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

(A2)


ϕ(t) ϕ(0) + δϕ(t)

ϕ(0) + 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns 1113946

ωt

o



Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns



Szωt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A3)


δvx(t) δvx(0) + 1113946t


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

(A4)



0ϕydt

δvy(t) δvy(0) + g 1113946t

0ϕxdt

(A5)



0ϕydt


01113874ϕy(0) + C

ns (2 1)



+ Cns (2 3)Szωt1113875dt



ω+

Mxz

ω+

Myz sinωt

ωminus Myzt1113888 1113889

minus Cns (2 2)g1113888

minus Mxz sinωt

ω+

Myz

ωminus

Myz cosωt

ω


g

2Szωt

2

1113874δvx(0) minus Cns (2 1)

g


+ Mxz1113875 + Cns (2 2)

g




ns (2 3)

g

2Szωt

21113875

(A6)


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

1113874δvy(0) + Cns (1 1)

g


+ Mxz1113875 minus Cns (1 2)

g




ns (1 3)

g

2Szωt

21113875

(A7)

Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018








δωbib asymp

Sx Mxy Mxz

Myx Sy Myz

Mzx Mzy Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ωbiex

ωbiey

ωbiez + ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

asymp

Sx Mxy Mxz

Myx Sy Myz

Mzx Mzy Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ω middot

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(3)


Cnb C

ns middot C

sb (4)

Cns




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Csb


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)




ϕ(t) ϕ(0) + δϕ(t)

ϕ(0) + Cns



Szωt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(6)






δvx(t) δvx(0) + 1113946t


1113874δvx(0) minus Cns (2 1)

g


+ Myz sinωt + Mxz1113875 + Cns (2 2)

g

ω1113874Mxz sinωt



ns (2 2)gMxzt

minus Cns (2 3)

g

2Szωt

21113875

(8)




δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

δvy(0) + Cns (1 1)

g


ns (1 2)

g



ns (1 2)gMxzt + C

ns (1 3)

g

2Szωt

21113874 1113875

(9)






(10)



_Mxz 0

_Myz 0

_Sz 0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(11)


prime



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

ω ρ +Ω

(12)






δ _vz 2g


+fxψy + nablaz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)



δ _θx minusvz

Rδθx minus

1Rδvy

δ _θy minusvz

Rδθy +

1Rδvx

δ _h δvz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)


εx

εy

εz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

n

Cnb

Bx

By

Bz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

Sx Mxy Mxz

Myx Sy Myz

Mzx Mzy Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦middot

ωxib

ωy

ib

ωzib

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(15)






Z1 δV Vins minus 0

δvx

δvy

δvz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)


Z1(t) H1X(t) + η (17)


H1




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (18)


1113957Cb





ib minus 1113957Cb


eb

δ1113957ωbib + C

bn ωn

ie times( 1113857ϕn(20)



ie

ωbibz 0 0

0 ωbibz 0

0 0 ωbibz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21)


4 Simulation





IMU(gyros accelero


Zero velocity


Estimated errors


Mxz Myz Sz

Mxz Myz Sz









005 005 005 20 60 20 60 20 20 100 100 300 002



Estimate 5996 5996 30002



100 100 100 20 20 20 20 20 20 100 100 100 30

0 50 100 150 200 250 300 350 4000

50

100

150

Time (s)

S z (p

pm)

(a)

0 50 100 150 200 250 300 350 4000

5

10

15

20

Time (s)

Mxz

(arc

-sec

)

(b)

0 50 100 150 200 250 300 350 4000

5

10

15

20

Time (s)

Myz

(arc

-sec

)

(c)








5 Real Test




0 50 100 150 200 250 300 350 4000

100

200

300

400

Time (s)

S z (p

pm)

(a)

0 50 100 150 200 250 300 350 4000

20

40

60

80

Time (s)

Mxz

(arc

-sec

)

(b)

0 50 100 150 200 250 300 350 4000

20

40

60

80

Time (s)

Myz

(arc

-sec

)

(c)








0 200 400 600 800 1000 1200ndash2

0

2

4

6

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash3

ndash2

ndash1

0

1

Time (s)

V y (m

s)

(b)




Estimated 1542511 minus 1077430 840126



0 50 100 150 200 250 300 350 400

1

0

2

3

Time (s)

V x (m

s)

(a)

0 50 100 150 200 250 300 350 400ndash10

0

10

20

Time (s)

V y (m

s)

(b)




6 Conclusions


0 200 400 600 800 1000 1200ndash02

0

02

04

06

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash1

ndash05

0

05

Time (s)

V y (m

s)

(b)



Parameter eoryvalue

Beforecompensation

Aftercompensation



0 200 400 600 800 1000 12000

50

100

Time (s)

S z (p

pm)

(a)

0 200 400 600 800 1000 12000

100

200

Time (s)

Mxz

(arc

-sec

)

(b)

0 200 400 600 800 1000 1200ndash200

ndash100

0

Time (s)

Myz

(arc

-sec

)

(c)





Appendix


ϕ(t) ϕ(0) + δϕ(t) (A1)


δϕ(t) 1113946t

0C

nbδω

bibdt

asymp 1113946t

0C

nb

minus minus Mxz

minus minus Myz

minus minus Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dt 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

(A2)


ϕ(t) ϕ(0) + δϕ(t)

ϕ(0) + 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns 1113946

ωt

o



Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns



Szωt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A3)


δvx(t) δvx(0) + 1113946t


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

(A4)



0ϕydt

δvy(t) δvy(0) + g 1113946t

0ϕxdt

(A5)



0ϕydt


01113874ϕy(0) + C

ns (2 1)



+ Cns (2 3)Szωt1113875dt



ω+

Mxz

ω+

Myz sinωt

ωminus Myzt1113888 1113889

minus Cns (2 2)g1113888

minus Mxz sinωt

ω+

Myz

ωminus

Myz cosωt

ω


g

2Szωt

2

1113874δvx(0) minus Cns (2 1)

g


+ Mxz1113875 + Cns (2 2)

g




ns (2 3)

g

2Szωt

21113875

(A6)


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

1113874δvy(0) + Cns (1 1)

g


+ Mxz1113875 minus Cns (1 2)

g




ns (1 3)

g

2Szωt

21113875

(A7)

Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018








δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

δvy(0) + Cns (1 1)

g


ns (1 2)

g



ns (1 2)gMxzt + C

ns (1 3)

g

2Szωt

21113874 1113875

(9)






(10)



_Mxz 0

_Myz 0

_Sz 0

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(11)


prime



⎧⎪⎪⎪⎨

⎪⎪⎪⎩

ω ρ +Ω

(12)






δ _vz 2g


+fxψy + nablaz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(13)



δ _θx minusvz

Rδθx minus

1Rδvy

δ _θy minusvz

Rδθy +

1Rδvx

δ _h δvz

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)


εx

εy

εz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

n

Cnb

Bx

By

Bz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

Sx Mxy Mxz

Myx Sy Myz

Mzx Mzy Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦middot

ωxib

ωy

ib

ωzib

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(15)






Z1 δV Vins minus 0

δvx

δvy

δvz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)


Z1(t) H1X(t) + η (17)


H1




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (18)


1113957Cb





ib minus 1113957Cb


eb

δ1113957ωbib + C

bn ωn

ie times( 1113857ϕn(20)



ie

ωbibz 0 0

0 ωbibz 0

0 0 ωbibz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21)


4 Simulation





IMU(gyros accelero


Zero velocity


Estimated errors


Mxz Myz Sz

Mxz Myz Sz









005 005 005 20 60 20 60 20 20 100 100 300 002



Estimate 5996 5996 30002



100 100 100 20 20 20 20 20 20 100 100 100 30

0 50 100 150 200 250 300 350 4000

50

100

150

Time (s)

S z (p

pm)

(a)

0 50 100 150 200 250 300 350 4000

5

10

15

20

Time (s)

Mxz

(arc

-sec

)

(b)

0 50 100 150 200 250 300 350 4000

5

10

15

20

Time (s)

Myz

(arc

-sec

)

(c)








5 Real Test




0 50 100 150 200 250 300 350 4000

100

200

300

400

Time (s)

S z (p

pm)

(a)

0 50 100 150 200 250 300 350 4000

20

40

60

80

Time (s)

Mxz

(arc

-sec

)

(b)

0 50 100 150 200 250 300 350 4000

20

40

60

80

Time (s)

Myz

(arc

-sec

)

(c)








0 200 400 600 800 1000 1200ndash2

0

2

4

6

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash3

ndash2

ndash1

0

1

Time (s)

V y (m

s)

(b)




Estimated 1542511 minus 1077430 840126



0 50 100 150 200 250 300 350 400

1

0

2

3

Time (s)

V x (m

s)

(a)

0 50 100 150 200 250 300 350 400ndash10

0

10

20

Time (s)

V y (m

s)

(b)




6 Conclusions


0 200 400 600 800 1000 1200ndash02

0

02

04

06

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash1

ndash05

0

05

Time (s)

V y (m

s)

(b)



Parameter eoryvalue

Beforecompensation

Aftercompensation



0 200 400 600 800 1000 12000

50

100

Time (s)

S z (p

pm)

(a)

0 200 400 600 800 1000 12000

100

200

Time (s)

Mxz

(arc

-sec

)

(b)

0 200 400 600 800 1000 1200ndash200

ndash100

0

Time (s)

Myz

(arc

-sec

)

(c)





Appendix


ϕ(t) ϕ(0) + δϕ(t) (A1)


δϕ(t) 1113946t

0C

nbδω

bibdt

asymp 1113946t

0C

nb

minus minus Mxz

minus minus Myz

minus minus Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dt 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

(A2)


ϕ(t) ϕ(0) + δϕ(t)

ϕ(0) + 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns 1113946

ωt

o



Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns



Szωt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A3)


δvx(t) δvx(0) + 1113946t


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

(A4)



0ϕydt

δvy(t) δvy(0) + g 1113946t

0ϕxdt

(A5)



0ϕydt


01113874ϕy(0) + C

ns (2 1)



+ Cns (2 3)Szωt1113875dt



ω+

Mxz

ω+

Myz sinωt

ωminus Myzt1113888 1113889

minus Cns (2 2)g1113888

minus Mxz sinωt

ω+

Myz

ωminus

Myz cosωt

ω


g

2Szωt

2

1113874δvx(0) minus Cns (2 1)

g


+ Mxz1113875 + Cns (2 2)

g




ns (2 3)

g

2Szωt

21113875

(A6)


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

1113874δvy(0) + Cns (1 1)

g


+ Mxz1113875 minus Cns (1 2)

g




ns (1 3)

g

2Szωt

21113875

(A7)

Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018










Z1 δV Vins minus 0

δvx

δvy

δvz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (16)


Z1(t) H1X(t) + η (17)


H1




⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (18)


1113957Cb





ib minus 1113957Cb


eb

δ1113957ωbib + C

bn ωn

ie times( 1113857ϕn(20)



ie

ωbibz 0 0

0 ωbibz 0

0 0 ωbibz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21)


4 Simulation





IMU(gyros accelero


Zero velocity


Estimated errors


Mxz Myz Sz

Mxz Myz Sz









005 005 005 20 60 20 60 20 20 100 100 300 002



Estimate 5996 5996 30002



100 100 100 20 20 20 20 20 20 100 100 100 30

0 50 100 150 200 250 300 350 4000

50

100

150

Time (s)

S z (p

pm)

(a)

0 50 100 150 200 250 300 350 4000

5

10

15

20

Time (s)

Mxz

(arc

-sec

)

(b)

0 50 100 150 200 250 300 350 4000

5

10

15

20

Time (s)

Myz

(arc

-sec

)

(c)








5 Real Test




0 50 100 150 200 250 300 350 4000

100

200

300

400

Time (s)

S z (p

pm)

(a)

0 50 100 150 200 250 300 350 4000

20

40

60

80

Time (s)

Mxz

(arc

-sec

)

(b)

0 50 100 150 200 250 300 350 4000

20

40

60

80

Time (s)

Myz

(arc

-sec

)

(c)








0 200 400 600 800 1000 1200ndash2

0

2

4

6

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash3

ndash2

ndash1

0

1

Time (s)

V y (m

s)

(b)




Estimated 1542511 minus 1077430 840126



0 50 100 150 200 250 300 350 400

1

0

2

3

Time (s)

V x (m

s)

(a)

0 50 100 150 200 250 300 350 400ndash10

0

10

20

Time (s)

V y (m

s)

(b)




6 Conclusions


0 200 400 600 800 1000 1200ndash02

0

02

04

06

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash1

ndash05

0

05

Time (s)

V y (m

s)

(b)



Parameter eoryvalue

Beforecompensation

Aftercompensation



0 200 400 600 800 1000 12000

50

100

Time (s)

S z (p

pm)

(a)

0 200 400 600 800 1000 12000

100

200

Time (s)

Mxz

(arc

-sec

)

(b)

0 200 400 600 800 1000 1200ndash200

ndash100

0

Time (s)

Myz

(arc

-sec

)

(c)





Appendix


ϕ(t) ϕ(0) + δϕ(t) (A1)


δϕ(t) 1113946t

0C

nbδω

bibdt

asymp 1113946t

0C

nb

minus minus Mxz

minus minus Myz

minus minus Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dt 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

(A2)


ϕ(t) ϕ(0) + δϕ(t)

ϕ(0) + 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns 1113946

ωt

o



Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns



Szωt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A3)


δvx(t) δvx(0) + 1113946t


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

(A4)



0ϕydt

δvy(t) δvy(0) + g 1113946t

0ϕxdt

(A5)



0ϕydt


01113874ϕy(0) + C

ns (2 1)



+ Cns (2 3)Szωt1113875dt



ω+

Mxz

ω+

Myz sinωt

ωminus Myzt1113888 1113889

minus Cns (2 2)g1113888

minus Mxz sinωt

ω+

Myz

ωminus

Myz cosωt

ω


g

2Szωt

2

1113874δvx(0) minus Cns (2 1)

g


+ Mxz1113875 + Cns (2 2)

g




ns (2 3)

g

2Szωt

21113875

(A6)


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

1113874δvy(0) + Cns (1 1)

g


+ Mxz1113875 minus Cns (1 2)

g




ns (1 3)

g

2Szωt

21113875

(A7)

Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018














005 005 005 20 60 20 60 20 20 100 100 300 002



Estimate 5996 5996 30002



100 100 100 20 20 20 20 20 20 100 100 100 30

0 50 100 150 200 250 300 350 4000

50

100

150

Time (s)

S z (p

pm)

(a)

0 50 100 150 200 250 300 350 4000

5

10

15

20

Time (s)

Mxz

(arc

-sec

)

(b)

0 50 100 150 200 250 300 350 4000

5

10

15

20

Time (s)

Myz

(arc

-sec

)

(c)








5 Real Test




0 50 100 150 200 250 300 350 4000

100

200

300

400

Time (s)

S z (p

pm)

(a)

0 50 100 150 200 250 300 350 4000

20

40

60

80

Time (s)

Mxz

(arc

-sec

)

(b)

0 50 100 150 200 250 300 350 4000

20

40

60

80

Time (s)

Myz

(arc

-sec

)

(c)








0 200 400 600 800 1000 1200ndash2

0

2

4

6

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash3

ndash2

ndash1

0

1

Time (s)

V y (m

s)

(b)




Estimated 1542511 minus 1077430 840126



0 50 100 150 200 250 300 350 400

1

0

2

3

Time (s)

V x (m

s)

(a)

0 50 100 150 200 250 300 350 400ndash10

0

10

20

Time (s)

V y (m

s)

(b)




6 Conclusions


0 200 400 600 800 1000 1200ndash02

0

02

04

06

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash1

ndash05

0

05

Time (s)

V y (m

s)

(b)



Parameter eoryvalue

Beforecompensation

Aftercompensation



0 200 400 600 800 1000 12000

50

100

Time (s)

S z (p

pm)

(a)

0 200 400 600 800 1000 12000

100

200

Time (s)

Mxz

(arc

-sec

)

(b)

0 200 400 600 800 1000 1200ndash200

ndash100

0

Time (s)

Myz

(arc

-sec

)

(c)





Appendix


ϕ(t) ϕ(0) + δϕ(t) (A1)


δϕ(t) 1113946t

0C

nbδω

bibdt

asymp 1113946t

0C

nb

minus minus Mxz

minus minus Myz

minus minus Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dt 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

(A2)


ϕ(t) ϕ(0) + δϕ(t)

ϕ(0) + 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns 1113946

ωt

o



Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns



Szωt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A3)


δvx(t) δvx(0) + 1113946t


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

(A4)



0ϕydt

δvy(t) δvy(0) + g 1113946t

0ϕxdt

(A5)



0ϕydt


01113874ϕy(0) + C

ns (2 1)



+ Cns (2 3)Szωt1113875dt



ω+

Mxz

ω+

Myz sinωt

ωminus Myzt1113888 1113889

minus Cns (2 2)g1113888

minus Mxz sinωt

ω+

Myz

ωminus

Myz cosωt

ω


g

2Szωt

2

1113874δvx(0) minus Cns (2 1)

g


+ Mxz1113875 + Cns (2 2)

g




ns (2 3)

g

2Szωt

21113875

(A6)


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

1113874δvy(0) + Cns (1 1)

g


+ Mxz1113875 minus Cns (1 2)

g




ns (1 3)

g

2Szωt

21113875

(A7)

Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018













5 Real Test




0 50 100 150 200 250 300 350 4000

100

200

300

400

Time (s)

S z (p

pm)

(a)

0 50 100 150 200 250 300 350 4000

20

40

60

80

Time (s)

Mxz

(arc

-sec

)

(b)

0 50 100 150 200 250 300 350 4000

20

40

60

80

Time (s)

Myz

(arc

-sec

)

(c)








0 200 400 600 800 1000 1200ndash2

0

2

4

6

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash3

ndash2

ndash1

0

1

Time (s)

V y (m

s)

(b)




Estimated 1542511 minus 1077430 840126



0 50 100 150 200 250 300 350 400

1

0

2

3

Time (s)

V x (m

s)

(a)

0 50 100 150 200 250 300 350 400ndash10

0

10

20

Time (s)

V y (m

s)

(b)




6 Conclusions


0 200 400 600 800 1000 1200ndash02

0

02

04

06

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash1

ndash05

0

05

Time (s)

V y (m

s)

(b)



Parameter eoryvalue

Beforecompensation

Aftercompensation



0 200 400 600 800 1000 12000

50

100

Time (s)

S z (p

pm)

(a)

0 200 400 600 800 1000 12000

100

200

Time (s)

Mxz

(arc

-sec

)

(b)

0 200 400 600 800 1000 1200ndash200

ndash100

0

Time (s)

Myz

(arc

-sec

)

(c)





Appendix


ϕ(t) ϕ(0) + δϕ(t) (A1)


δϕ(t) 1113946t

0C

nbδω

bibdt

asymp 1113946t

0C

nb

minus minus Mxz

minus minus Myz

minus minus Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dt 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

(A2)


ϕ(t) ϕ(0) + δϕ(t)

ϕ(0) + 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns 1113946

ωt

o



Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns



Szωt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A3)


δvx(t) δvx(0) + 1113946t


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

(A4)



0ϕydt

δvy(t) δvy(0) + g 1113946t

0ϕxdt

(A5)



0ϕydt


01113874ϕy(0) + C

ns (2 1)



+ Cns (2 3)Szωt1113875dt



ω+

Mxz

ω+

Myz sinωt

ωminus Myzt1113888 1113889

minus Cns (2 2)g1113888

minus Mxz sinωt

ω+

Myz

ωminus

Myz cosωt

ω


g

2Szωt

2

1113874δvx(0) minus Cns (2 1)

g


+ Mxz1113875 + Cns (2 2)

g




ns (2 3)

g

2Szωt

21113875

(A6)


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

1113874δvy(0) + Cns (1 1)

g


+ Mxz1113875 minus Cns (1 2)

g




ns (1 3)

g

2Szωt

21113875

(A7)

Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018













0 200 400 600 800 1000 1200ndash2

0

2

4

6

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash3

ndash2

ndash1

0

1

Time (s)

V y (m

s)

(b)




Estimated 1542511 minus 1077430 840126



0 50 100 150 200 250 300 350 400

1

0

2

3

Time (s)

V x (m

s)

(a)

0 50 100 150 200 250 300 350 400ndash10

0

10

20

Time (s)

V y (m

s)

(b)




6 Conclusions


0 200 400 600 800 1000 1200ndash02

0

02

04

06

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash1

ndash05

0

05

Time (s)

V y (m

s)

(b)



Parameter eoryvalue

Beforecompensation

Aftercompensation



0 200 400 600 800 1000 12000

50

100

Time (s)

S z (p

pm)

(a)

0 200 400 600 800 1000 12000

100

200

Time (s)

Mxz

(arc

-sec

)

(b)

0 200 400 600 800 1000 1200ndash200

ndash100

0

Time (s)

Myz

(arc

-sec

)

(c)





Appendix


ϕ(t) ϕ(0) + δϕ(t) (A1)


δϕ(t) 1113946t

0C

nbδω

bibdt

asymp 1113946t

0C

nb

minus minus Mxz

minus minus Myz

minus minus Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dt 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

(A2)


ϕ(t) ϕ(0) + δϕ(t)

ϕ(0) + 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns 1113946

ωt

o



Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns



Szωt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A3)


δvx(t) δvx(0) + 1113946t


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

(A4)



0ϕydt

δvy(t) δvy(0) + g 1113946t

0ϕxdt

(A5)



0ϕydt


01113874ϕy(0) + C

ns (2 1)



+ Cns (2 3)Szωt1113875dt



ω+

Mxz

ω+

Myz sinωt

ωminus Myzt1113888 1113889

minus Cns (2 2)g1113888

minus Mxz sinωt

ω+

Myz

ωminus

Myz cosωt

ω


g

2Szωt

2

1113874δvx(0) minus Cns (2 1)

g


+ Mxz1113875 + Cns (2 2)

g




ns (2 3)

g

2Szωt

21113875

(A6)


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

1113874δvy(0) + Cns (1 1)

g


+ Mxz1113875 minus Cns (1 2)

g




ns (1 3)

g

2Szωt

21113875

(A7)

Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018









6 Conclusions


0 200 400 600 800 1000 1200ndash02

0

02

04

06

Time (s)

V x (m

s)

(a)

0 200 400 600 800 1000 1200ndash1

ndash05

0

05

Time (s)

V y (m

s)

(b)



Parameter eoryvalue

Beforecompensation

Aftercompensation



0 200 400 600 800 1000 12000

50

100

Time (s)

S z (p

pm)

(a)

0 200 400 600 800 1000 12000

100

200

Time (s)

Mxz

(arc

-sec

)

(b)

0 200 400 600 800 1000 1200ndash200

ndash100

0

Time (s)

Myz

(arc

-sec

)

(c)





Appendix


ϕ(t) ϕ(0) + δϕ(t) (A1)


δϕ(t) 1113946t

0C

nbδω

bibdt

asymp 1113946t

0C

nb

minus minus Mxz

minus minus Myz

minus minus Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dt 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

(A2)


ϕ(t) ϕ(0) + δϕ(t)

ϕ(0) + 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns 1113946

ωt

o



Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns



Szωt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A3)


δvx(t) δvx(0) + 1113946t


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

(A4)



0ϕydt

δvy(t) δvy(0) + g 1113946t

0ϕxdt

(A5)



0ϕydt


01113874ϕy(0) + C

ns (2 1)



+ Cns (2 3)Szωt1113875dt



ω+

Mxz

ω+

Myz sinωt

ωminus Myzt1113888 1113889

minus Cns (2 2)g1113888

minus Mxz sinωt

ω+

Myz

ωminus

Myz cosωt

ω


g

2Szωt

2

1113874δvx(0) minus Cns (2 1)

g


+ Mxz1113875 + Cns (2 2)

g




ns (2 3)

g

2Szωt

21113875

(A6)


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

1113874δvy(0) + Cns (1 1)

g


+ Mxz1113875 minus Cns (1 2)

g




ns (1 3)

g

2Szωt

21113875

(A7)

Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018










Appendix


ϕ(t) ϕ(0) + δϕ(t) (A1)


δϕ(t) 1113946t

0C

nbδω

bibdt

asymp 1113946t

0C

nb

minus minus Mxz

minus minus Myz

minus minus Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

0

0

ω

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dt 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

(A2)


ϕ(t) ϕ(0) + δϕ(t)

ϕ(0) + 1113946ωt

0C

nb

Mxz

Myz

Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns 1113946

ωt

o



Sz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦dφ

ϕ(0) + Cns



Szωt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(A3)


δvx(t) δvx(0) + 1113946t


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

(A4)



0ϕydt

δvy(t) δvy(0) + g 1113946t

0ϕxdt

(A5)



0ϕydt


01113874ϕy(0) + C

ns (2 1)



+ Cns (2 3)Szωt1113875dt



ω+

Mxz

ω+

Myz sinωt

ωminus Myzt1113888 1113889

minus Cns (2 2)g1113888

minus Mxz sinωt

ω+

Myz

ωminus

Myz cosωt

ω


g

2Szωt

2

1113874δvx(0) minus Cns (2 1)

g


+ Mxz1113875 + Cns (2 2)

g




ns (2 3)

g

2Szωt

21113875

(A6)


δvy(t) δvy(0) + 1113946t

0δfy + gϕx1113872 1113873dt

1113874δvy(0) + Cns (1 1)

g


+ Mxz1113875 minus Cns (1 2)

g




ns (1 3)

g

2Szωt

21113875

(A7)

Data Availability





References




































Journal of












Journal of







Volume 2018






Volume 2018










References




































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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