Research ArticleA Preferable Airborne Integrated Navigation Method Based onINS and GPS

Xiaoyue Zhang and Kaiwen Ning

School of Instrumentation Science and Optoelectronics Engineering, Beihang University, Beijing 100191, China

Correspondence should be addressed to Xiaoyue Zhang; zhangxiaoyue@buaa.edu.cn

Received 24 August 2017; Revised 10 January 2018; Accepted 6 February 2018; Published 19 June 2018

Academic Editor: Eduard Llobet

Copyright © 2018 Xiaoyue Zhang and Kaiwen Ning. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original workis properly cited.

An integrated navigation method based on INS and GPS was proposed for airborne navigation. The influence of scale factor errorand misalignment error of gyroscope and accelerometer on navigation accuracy was analyzed. Compared with traditional INS/GPSintegrated navigation method, scale factor error and misalignment error were added to the state model of the integrated navigationsystem. The observability of scale factor error and misalignment error was analyzed combined with typical airborne movement.Then the integrated system was optimized, and the new navigation model of the integrated system was obtained. The optimizedINS/GPS integrated model was validated by numerical simulation and turntable test. Comparing the proposed model withtraditional integrated model (integrated system error states do not include scale factor error and misalignment error), the resultsshowed that the proposed integrated navigation method can improve the accuracy from 8% to 28% of the east, north, andupward positions.

1. Introduction

Inertial navigation system (INS) is an autonomous naviga-tion system that does not depend on any external informa-tion [1, 2]. However, the characteristics of the locationerror accumulate with time, making it difficult to workindependently for a long time. Global Positioning System(GPS) can measure three-dimensional position and velocityaccurately, but the disadvantage is susceptible to interfer-ence and control [3–5]. Therefore, INS and GPS havecomplementary characteristics. Since the 1990s, INS/GPSintegrated navigation system has been a great success athome and abroad, and it has developed into a specializedtechnology [6, 7].

INS/GPS integrated navigation system works as follows:when GPS signal is good, the system selects the integratednavigation mode. The precision of integrated navigationbasically depends on GPS precision, and the inertial mea-surement unit (IMU) errors can be estimated and compen-sated online. When GPS signal is disturbed or shielded, thesystem automatically shifts into inertial navigation mode.At this point, navigation accuracy basically depends on the

precision of IMU [8]. Therefore, the estimation accuracy ofIMU errors in integrated navigation can affect the accuracyof the subsequent inertial navigation [9, 10]. In airborneINS/GPS integrated navigation system, however, the errorsof IMU only consider the bias of gyroscope and accelerome-ter without considering scale factor and misalignment atpresent. A method of the dynamic parameter identificationof the scale factor error and misalignment error was designedbased on Kalman filter. The observability of the scale factorerror and misalignment error with different maneuvers wasanalyzed in [11]. Zhou et al. described the error dynamicsystem equation and observation equation of inertial naviga-tion system and the singular value of the system states ofonline calibration [12]. Therefore, a more advanced methodcan be designed to make the IMU errors (including bias, scalefactor, and misalignment) be more accurate in estimationand compensation in the integrated navigation process.When entering the inertial navigation mode, it can get highernavigation accuracy.

For the application of airborne navigation, this paperanalyzes the influence of scale factor error and misalignmenterror on the accuracy of integrated navigation. Based on the

HindawiJournal of SensorsVolume 2018, Article ID 7342896, 14 pageshttps://doi.org/10.1155/2018/7342896

analysis, the scale factor error and the misalignment error areadded to the error model of the integrated navigation system.Then the observability of the scale factor error and misalign-ment error is analyzed combined with the typical airbornemovement. According to the observability analysis results,the integrated system is optimized and the new error modelof the integrated navigation system is obtained. Finally, theoptimized INS/GPS integrated model proposed in this paperis validated by numerical simulation and turntable test, andthen the proposed model is compared with the traditionalintegrated model.

2. Error Analysis of Airborne InertialMeasurement Unit

The errors of inertial measurement unit mainly includebias, scale factor error and misalignment error. The errormodel of INS and IMU is given in the following passage,and the influence of IMU errors on the navigation accuracyis analyzed combined typical airborne movement.

2.1. Error Model of Inertial Navigation System.

δλ =vnE tan L sec L

RN + hδL −

vnE sec LRN + h 2 δh +

sec LRN + h

δvnE,

δL = −vnN

RM + h 2 δh +1

RM + hδvnN ,

δh = δvnU ,

δvn = fn × φ − 2ωnie + ωn

en × δvn + vn × 2δωnie + δωn

en + δfn,φ = − ωn

ie + ωnen × φ + δωn

ie + δωnen − εn,

1

where δλ, δL, and δh represent the longitude error, thelatitude error, and the altitude error, respectively, and RMand RN are the curvature radius of the meridian and primevertical. δvn and φ represent velocity error and attitude error,respectively, and δfn represents the accelerometer measure-ment error, which contains accelerometer bias, scale factorerror, and misalignment error. εn represents the gyroscopemeasurement error, which contains gyroscope bias, scalefactor error, and misalignment error.

2.2. Error Model of Inertial Measurement Unit.

εbx

εby

εbz

=

gBx

gBy

gBz

+

gSFx gMAxy gMAxz

gMAyx gSFy gMAyz

gMAzx gMAzy gSFz

ωbibx

ωbiby

ωbibz

,

2

where gSFx, gSFy, and gSFz are the gyroscope scale factorerrors of the x-axis, y-axis, and z-axis. gMAxy, gMAxz ,gMAyx, gMAyz , gMAzx, and gMAzy are the gyroscope

misalignment errors. gBx, gBy, and gBz are the gyroscope

biases. ωbibx, ω

biby , and ωb

ibz are the gyroscope ideal outputs.

δf bx

δf by

δf bz

=

aBx

aBy

aBz

+

aSFx aMAxy aMAxz

aMAyx aSFy aMAyz

aMAzx aMAzy aSFz

f bx

f by

f bz

,

3

where aSFx, aSFy, and aSFz are the accelerometer scalefactor errors of the x-axis, y-axis, and z-axis. aMAxy ,aMAxz , aMAyx, aMAyz , aMAzx, and aMAzy are the acceler-ometer misalignment errors. aBx, aBy, and aBz are the

accelerometer biases. f bix , fby , and f bz are the accelerometer

ideal outputs [13].

2.3. Analysis of the Influence of IMU Errors on NavigationAccuracy. In order to facilitate quantitative analysis, com-bined with the actual low-precision inertial navigationsystems commonly used in airborne navigation, the errorparameters of the IMU are set as follows:

gB = 1°/h,

gSFx = gSFy = gSFz = 300 ppm,

gMAxy = gMAxz = gMAyx = gMAyz = gMAzx = gMAzy = 40″,aB = 500 μg,

aSFx = aSFy = aSFz = 300 ppm,

aMAxy = aMAxz = aMAyx = aMAyz = aMAzx = aMAzy = 40″4

2.3.1. Analysis of Attitude Error. In order to more clearly andeasily analyze the influence of gyroscope scale factor errorand misalignment error on attitude error, we temporarilydo not consider other terms. The attitude error equationcan be simplified as

φ = −Cnb

gSFx gMAxy gMAxz

gMAyx gSFy gMAyz

gMAzx gMAzy gSFz

ωbibx

ωbiby

ωbibz

+

gBx

gBy

gBz

= −Cnb δKGωb

ib + gB

5

When the airframe turns, the angular velocity of theEarth and the platform are small relative to the IMU rotationrate. So we ignore the influence of angular velocity of theEarth and the platform on the attitude error; the attitudeerror equation is further simplified [14]:

φ = −Cnb δKG ωb

in +ωbnb + gB ≈ −Cn

b δKGωbnb + gB

6

2 Journal of Sensors

When the system only changes the pitch angle, theroll angle and heading angle change could be assumedto be zero, then

ωbnb = ωx 0 0 T , Cn

b ≈

1 0 0

0 cos θ −sin θ

0 sin θ cos θ

7

Substituting (7) into (6),

φ = −ωx ⋅

gSFx

cos θ ⋅ gMAyx − sin θ ⋅ gMAzx

cos θ ⋅ gMAzx + sin θ ⋅ gMAyx

−

gBx

cos θ ⋅ gBy − sin θ ⋅ gBz

cos θ ⋅ gBz + sin θ ⋅ gBy

8

Combined with the gyroscope error parameters givenabove and (8), the influence of gyroscope scale factor errorgSFx on attitude error is greater than that of gyroscope biasgBx on attitude error when ωx > 0 93°/s; the influenceof gyroscope misalignment errors gMAyx and gMAzx onattitude error is greater than that of gyro bias gBx andgBz on attitude error when ωx > 1 43°/s. Therefore, whenthe body pitch angle changes, the influence of gyroscopescale factor error and misalignment error on attitude errorcannot be ignored.

Similarly, when the system roll angle changes or headingangle changes, then ωb

nb is

ωbnb = 0 ωy 0 T or ωb

nb = 0 0 ωzT 9

Substituting (9) into (6), we can get the same conclusion.

2.3.2. Error Analysis of Velocity and Position. The mostcommon movements of the aircraft are uniform motionand accelerated motion, this paper mainly analyzes theinfluence of the accelerometer errors on position error andvelocity error in these two kinds of motion. To simplify theanalysis, this paper also does not consider other error terms,so velocity error equation can be simplified as

δvn =Cnb

aSFx aMAxy aMAxz

aMAyx aSFy aMAyz

aMAzx aMAzy aSFz

f bx

f by

f bz

+

aBx

aBy

aBz

=Cnb δKA fbsf + aB

10

When the system only changes the pitch angle, the rollangle and heading angle change could be assumed to be zero;the attitude matrix Cn

b can be simplified as

Cnb =

1 0 0

0 cos θ −sin θ

0 sin θ cos θ

11

Substituting (11) into (10),

δvn =

δvnE

δvnN

δvnU

≈

aBx

cos θ ⋅ aBy − sin θ ⋅ aBz

cos θ ⋅ aBz + sin θ ⋅ aBy

+ f x ⋅

aSFx

cos θ ⋅ aMAyx − sin θ ⋅ aMAzx

sin θ ⋅ aMAyx + cos θ ⋅ aMAzx

+ f y ⋅

aMAxy

cos θ ⋅ aSFy − sin θ ⋅ aMAzy

sin θ ⋅ aSFy + cos θ ⋅ aMAzy

+ g ⋅

aMAxz

cos θ ⋅ aMAyz − sin θ ⋅ aSFz

sin θ ⋅ aMAyz + cos θ ⋅ aSFz12

In uniformmotion (f x = f y ≈ 0), according to the acceler-ometer error parameters and (12), we can conclude that aMA ⋅ g ≈ 194 μg < aB = 500 μg. The influence of accelerometermisalignment errors aMAxz and aMAyz on the east and northvelocity errors is about 40% of the bias aBx and aBy. Inaddition, aSFz ⋅ g ≈ 300 μg < aBz = 500 μg. The influence ofthe accelerometer scale factor error aSFz on upward velocityerror is about 60% of accelerometer bias aBz . Therefore,the influence of the accelerometer scale factor error andmisalignment error on velocity error cannot be ignoredin uniform motion.

In accelerated motion (f x ≠ 0, f y ≠ 0), according to theaccelerometer error parameters and (12), we can concludethat when f y > 1 67g and f z > 1 67g, the influence of theaccelerometer scale factor errors aSFy and aSFz on the northvelocity error and upward velocity error is greater than thatof accelerometer bias aBy and aBz .When f y > 2 57g andf z > 2 57g, the influence of the accelerometer misalign-ment errors aMAzy and aMAyz on the north velocity errorand upward velocity error is greater than that of acceler-ometer bias aBz and aBy. In the actual motion, theacceleration of aircraft is about 0.3g. According to theanalysis above, the influence of the accelerometer scale

3Journal of Sensors

factor error on velocity error is about 18% of the accelerom-eter bias and the influence of accelerometer misalignmenterror on velocity error is 12% of the accelerometer bias inaccelerated movement.

Similarly, when the system’s roll angle changes orheading angle changes, attitude matrix Cn

b is

Cnb ≈

cos γ 0 sin γ

0 1 0

−sin γ 0 cos γ

or Cnb ≈

cos ψ sin ψ 0

−sin ψ cos ψ 0

0 0 113

Substituting (13) into (10), we can get the sameconclusion.

In summary, the influence of scale factor error andmisalignment error of gyroscope and accelerometer onnavigation cannot be ignored.

3. Model Establishment

3.1. INS/GPS Integrated Navigation Model. From what hasbeen analyzed above, we can conclude that the influenceof scale factor error and misalignment error on naviga-tion cannot be ignored in airborne navigation. Therefore,the scale factor error and misalignment error need to beconsidered in the INS/GPS integrated error model. In thispaper, INS/GPS integrated navigation is realized by usingKalman filter. The position and velocity differences betweenGPS and INS are taken as measurement errors. IMU errorsand navigation errors can be online estimated and compen-sated, so the high navigation accuracy can be acquired.INS/GPS integrated navigation schematic diagram is shownin Figure 1.

3.1.1. Error State Model of Integrated Navigation. Theintegrated system error model can be expressed as

X t = F t X t +G t W t 14

X t is the 33-dimensional error vector of integratedsystem which can be expressed as

X t = δL δλ δh δvnE δvnN δvnU φE φN φU aBx aBy aBz gBx gBy

gBz aSFx aSFy aSFz aMAxy aMAxz aMAyx aMAyz

aMAzx aMAzy gSFx gSFy gSFz gMAxy gMAxz gMAyx

gMAyz gMAzx gMAzyT ,

15

W t is the system noise model which can be expressed as

W t = 01×3 aWx aWy aWz gWx gWy gWz 01×24T,

16

where aWx, aWy, and aWz are accelerometer noises, gWx ,gWy , and gWz are gyroscope noises.

G t is the noise driving matrix of integrated system,

G t =

03×3 03×3 03×303×3 Cn

b 03×303×3 03×3 Cn

b

09×24

024×9 024×24

17

3.1.2. Measurement Model of Integrated Navigation. Thevelocity and position differences between GPS and INS aretaken as measurement errors in the process of integratednavigation. So the measurement equation can be expressed as

Z t =Hp

HV

X t +Vp

VV

,

Hp =

0 R cos L 0⋮

R 0 0⋮ 03×300 0 0⋮

,

HV = 03×3 I3 03×27

18

VP and VV are the measurement noises of position andvelocity, respectively [15–17].

SINS

Error compensation,navigation calculation

Velocity, position,attitude

Sensor error

Data fusion

Velocity,positionNavigation

calculation

GPS receiver

Satellite basebandsignal processingRF front end

Gyroscope,accelerometer

Outputcorrection

information

Output the bestestimation

Figure 1: INS/GPS integrated navigation schematic.

4 Journal of Sensors

3.2. Airborne INS/GPS Integrated System Optimization. Scalefactor error and misalignment error are added in the newintegrated model. However, the scale factor error and mis-alignment error cannot be observed completely. It is difficultto obtain the desired estimation results if the estimated statescannot be observed. So the observability of the scale factorerror and misalignment error is analyzed by a combinedtypical airborne trajectory. Based on the analysis of observ-ability, we delete unobservable error states and get theoptimized integrated navigation model.

3.2.1. Observability Analysis. In this paper, the movementof aircraft is divided into the following three typical pro-cesses: straight flight, climbing flight, and turning flight.Straight flight includes uniform motion and acceleratedmotion. Climbing flight includes preparation for climbing,climbing, and transformation level. Turning flight includestilting and spiral. Flight trajectory consists of the threeprocesses above. Considering the complexity of the proce-dure, this paper does not simulate the whole flight pro-cess but chooses nine typical stages [18, 19]. The flightstages of aircraft and simulation trajectory are shown inTable 1 and Figure 2.

With the combined typical flight trajectory above,according to the eigenvalues of the error covariance matrix,

the observability of the scale factor error and misalignmenterror of the gyroscope and accelerometer is analyzed indifferent maneuvering conditions. And the influence ofdifferent maneuvering conditions on observability of eacherror term is analyzed. The eigenvalues of the covariancematrix can reflect whether the system state estimation isgood or bad. The smaller the eigenvalue is, the smallerthe estimated variance of the corresponding state is andthe better the degree of observability is. Otherwise, theestimation accuracy is low and the degree of observabilityis poor.

Figures 3 and 4 are the normalized eigenvalues of thecorresponding errors in the covariance matrix during thewhole motion. Tables 2 and 3 are the normalized covari-ance of scale error and misalignment error of gyroscopeand accelerometer.

As can be seen from Figure 3 and Table 2, the scale factorerror and misalignment error of gyroscope are not observableduring gliding. When climbing, the degree of observabilityof gSFx is increased. In a uniform flight, the flight time islong and the observability of gSFx and gMAyx is significantlyincreased. The degree of observability of gSFy and gMAyz isincreased when circling to the left and the degree of observ-ability of gMAxz is increased during circle exit. It can beseen that rotational maneuvering has a good incentive

Table 1: The flight stages of aircraft.

Flight stage Duration (s) Acceleration (m/s2) Yaw (°) Yaw rate (°/s) Pitch (°) Pitch rate (°/s) Roll (°) Roll rate (°/s)

Glide 20 2.5 45 0 0 0 0 0

Preparation for climbing 5 0 45 0 0 6 0 0

Climbing 100 0 45 0 30 0 0 0

Transformation level 5 0 45 0 30 −6 0 0

Uniform flight 200 0 45 0 0 0 0 0

Tilt to the left 20 0 45 0 0 0 0 1

Circle to the left 45 0 45 2 0 0 20 0

Exit circle 20 0 135 0 0 0 20 −1Uniform flight 285 0 135 0 0 0 0 0

39.9540

40.0540.1

116.3116.4

116.5116.6

116.7116.8

0

500

1000

1500

2000

2500

3000

Latitude (°)Longitude (°)

Alti

tude

(m)

Figure 2: The flight trajectory of aircraft.

5Journal of Sensors

for gSFy, gMAyz , and gMAxz . So gSFx, gSFy, gMAxz ,gMAyx, and gMAyz are observable. Moreover, at the endof the whole movement, the normalized covariance ofgSFz , gMAxy, gMAzx, and gMAzy are decreased not obviously.In summary, gSFz , gMAxy , gMAzx, and gMAzy of gyroscopeerrors are unobservable.

As can be seen from Figure 4 and Table 3, thedegree of observability of scale factor error aSFz isincreased during accelerated gliding. The change of theobservability degree of scale factor error and misalignmenterror is not obvious during climbing and transformationlevel. In a uniform flight, the degree of observability of

0.7

0.8

0.9

1

gSF x

5000 300 400100 200t (s)

(a)

0.7

0.8

0.9

1

gSF y

100 200 300 400 5000t (s)

(b)

0.94

0.96

0.98

1

gSF z

100 200 300 400 5000t (s)

(c)

100 200 300 400 5000t (s)

0.9

0.92

0.94

0.96

0.98

1

gMAxy

(d)

0 100 200 300 400 5000.8

0.85

0.9

0.95

1

t (s)

gMAxz

(e)

0.8

0.85

0.9

0.95

1

gMAyx

100 200 300 400 5000t (s)

(f)

100 200 300 400 5000t (s)

0.75

0.8

0.85

0.9

0.95

1

gMAyz

(g)

100 200 300 400 5000t (s)

0.98

0.985

0.99

0.995

1

gMAzx

(h)

0.999

0.9992

0.9994

0.9996

0.9998

1

gMAzy

100 200 300 400 5000t (s)

(i)

Figure 3: Eigenvalues of covariance matrix corresponding to gyroscope errors.

6 Journal of Sensors

aSFy and aMAxy is significantly increased. The degree ofobservability of aMAzx is increased during circling to theleft and exit circle. So aSFy, aSFz , aMAxy, and aMAzx areobservable. Moreover, at the end of the whole movement,the normalized covariance of aSFx, aMAxz , aMAyx, aMAyz ,and aMAzy are decreased not obviously. In summary, aSFx,

aMAxz , aMAyx , aMAyz , and aMAzy of accelerometer errorsare unobservable.

3.2.2. Optimization of Integrated System. As can be seenfrom the observability analysis results, the error statesgSFz , gMAxy, gMAzx, gMAzy, aSFx, aMAxz , aMAyx, aMAyz ,

0.95

0.96

0.97

0.98

0.99

1

aSF x

100 200 300 400 5000t (s)

(a)

0.7

0.8

0.9

1

aSF y

100 200 300 400 5000t (s)

(b)

0.75

0.8

0.85

0.9

0.95

1

aSF z

100 200 300 400 5000t (s)

(c)

0.8

0.85

0.9

0.95

1

aMAxy

100 200 300 400 5000t (s)

(d)

0.9

0.92

0.94

0.96

0.98

1

aMAxz

100 200 300 400 5000t (s)

(e)

0.96

0.97

0.98

0.99

1

aMAyx

100 200 300 400 5000t (s)

(f)

0.9

0.92

0.94

0.96

0.98

1

aMAyz

100 200 300 400 5000t (s)

(g)

0.8

0.85

0.9

0.95

1

aMAzx

100 200 300 400 5000t (s)

(h)

0.95

0.96

0.97

0.98

0.99

1

aMAzy

100 200 300 400 5000t (s)

(i)

Figure 4: Eigenvalues of covariance matrix corresponding to accelerometer errors.

7Journal of Sensors

and aMAzy are unobservable. After the unobservable errorsare deleted, the dimensions of the error states is reduced from33 to 24 and the optimized error state vector of the integratedsystem is

X t = δL δλ δh δvnE δvnN δvnU φE φN φU aBx aBy aBz gBx gBy

gBz aSFy aSFz aMAxy aMAzx gSFx gSFy gMAxz

gMAyx gMAyzT

19

So the system noise vector and the noise driving matrix ofthe integrated system are

W t = 01×3 aWx aWy aWz gWx gWy gWz 01×15 ,

G t =

03×3 03×3 03×303×3 Cn

b 03×3 09×1503×3 03×3 Cn

b

015×9 015×15

20

The measurement matrix is expressed as

Hp =

0 R cos L 0⋮

R 0 0⋮ 03×210 0 0⋮

,

Hv = 03×3 I3 03×18

21

In this way, the optimized integrated model is obtained,and then the simulation and turntable test are used to verifythe optimized model, respectively.

4. Model Verification

4.1. Simulation Verification. In this paper, numerical simula-tion is used to verify the validity and correctness of the pro-posed INS/GPS integrated model and then compare it withtraditional integrated model.

The error parameters of IMU and GPS are as follows:

(1) Gyroscope

(i) Bias: 1°/h; noise: 0 05°/ h

Table 3: Normalized covariance of accelerometer scale factor and misalignment.

Error state Initial value GlidePreparationfor climbing

ClimbingTransformation

levelUniformflight

Tilt to the leftCircle tothe left

Exit circle Final value

aSFx 1.00 1.00 1.00 1.00 1.00 1. 00 0.99 0.98 0.95 0.95

aSFy 1.00 0.99 0.99 0.95 0.95 0.75 0.73 0.69 0.69 0.69

aSFz 1.00 0.91 0.88 0.85 0.84 0.80 0.80 0.80 0.79 0.79

aMAxy 1.00 0.99 0.99 0.99 0.99 0.85 0.84 0.83 0.83 0.83

aMAxz 1.00 0.97 0.96 0.96 0.96 0.94 0.94 0.92 0.92 0.92

aMAyx 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.98 0.97 0.97

aMAyz 1.00 0.97 0.96 0.95 0.95 0.92 0.92 0.92 0.92 0.92

aMAzx 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.93 0.88 0.84

aMAzy 1.00 1.00 1.00 1.00 1.00 0.98 0.98 0.97 0.96 0.95

Table 2: Normalized covariance of gyroscope scale factor and misalignment.

Error state Initial value GlidePreparationfor climbing

ClimbingTransformation

levelUniformflight

Tilt tothe left

Circle tothe left

Exit circle Final value

gSFx 1.00 1.00 0.99 0.91 0.91 0.72 0.72 0.70 0.69 0.69

gSFy 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.88 0.87 0.71

gSFz 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.97 0.95

gMAxy 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.94 0.94 0.92

gMAxz 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.91 0.86 0.85

gMAyx 1.00 1.00 0.99 0.96 0.96 0.83 0.82 0.80 0.80 0.80

gMAyz 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.88 0.80 0.76

gMAzx 1.00 1.00 1.00 0.99 0.99 0.98 0.98 0.98 0.98 0.98

gMAzy 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99 0.99 0.99

8 Journal of Sensors

(ii) Scale factor error: 300 ppm; misalignment error:40″

(2) Accelerometer

(i) Bias: 500μg; noise: 100 μg/ Hz

(ii) Scale factor error: 300 ppm; misalignment error:40″

(3) GPS Receiver

(i) Position error: level 3m, vertical 5m

(ii) Velocity error: level 0.05m/s, vertical 0.05m/s

The simulation time is 700 seconds, the first 500 secondsfor INS/GPS integrated navigation. GPS failure after 500seconds, the receiver cannot output velocity and positioninformation. At this time, integrated system cannot updatemeasurement equation and turns into inertial navigationmode. The simulation results are as follows, Figure 5 showsthe estimation of gyroscope bias, scale factor error, andmisalignment error in integrated navigation. Figure 6 showsthe estimation of accelerometer bias, scale factor error, andmisalignment error in integrated navigation. Figure 7 showsthe true value, estimation of position and position error ofthe traditional integrated navigation model, and the pro-posed navigation model in the whole simulation process,including the east position, the north position, and theupward position.

From the simulation results above, the set errors of gyro-scope bias, scale factor, and misalignment are 1°/h, 300 ppm,and 40″, respectively. Figure 3 shows that the error estima-tion of the gyroscope are 0.96°/h, 305 ppm, and 42.8″, respec-tively. The set errors of accelerometer bias, scale factor, andmisalignment are 500μg, 300 ppm, and 40″, respectively.Figure 4 shows that the error estimation of the accelerometerare 482μg, 297 ppm, and 43″, respectively. Therefore, theobservable errors of IMU are effectively estimated in inte-grated navigation.

GPS signal is lost after 500 seconds; system conducted200 seconds inertial navigation at 700 seconds. Figure 7shows that the east, north, and upward position errors ofthe traditional model are 380m, 231.4m, and 126.1m,respectively. The east, north, and upward position errors ofthe proposed model are 271.2m, 175.1m, and 111.5m,respectively. The navigation accuracy of the east, north, andupward positions of the proposed model improved by28.6%, 24.3%, and 8.3%, respectively.

4.2. Turntable Test. Numerical simulation has proved thevalidity of the proposed integrated navigation method. Inorder to further verify the effectiveness of the method inactual integrated navigation system, turntable test is designedon the basis of numerical simulation. Test equipment includeINS/GPS integrated navigation system, turntable, two-wayDC power supply (0~30V, 0~3A), and data acquisition

gBxgBygBz

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Estim

atio

n of

gyr

osco

pe b

ias (

°/h)

1000 300 400200 500t (s)

(a)

gSFxgSFy

100 200 300 4000 500t (s)

−50

0

50

100

150

200

250

300

350

Estim

atio

n of

gyr

osco

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ale f

acto

r (pp

m)

(b)

gMAxzgMAyxgMAyz

100 200 300 400 5000t (s)

−10

0

10

20

30

40

50

60

Estim

atio

n of

gyr

osco

pe m

isalig

nmen

t (”)

(c)

Figure 5: Estimation of gyroscope errors. (a) Estimation ofgyroscope bias. (b) Estimation of gyroscope scale factor.(c) Estimation of gyroscope misalignment.

9Journal of Sensors

computer. Turntable is one of the important equipment intest for simulating the change of attitude and attitude rateof aircraft in space. INS/GPS integrated navigation systemconsists of inertial navigation system and GPS receiver. Theinstallation of the main test equipment is shown inFigures 8 and 9.

The error values of IMU are shown in Table 4.Based on the analysis of the typical airborne trajectory,

turntable is used to simulate the typical airborne movement.The rotation parameter of the turntable during the test isshown in Table 5.

The simulation time is 700 seconds, the first 500 secondsfor INS/GPS integrated navigation. Integrated system turnsinto inertial navigation mode after 500 seconds. The testresults are as follows, Figure 10 shows the estimation ofgyroscope bias, scale factor error, and misalignment errorin integrated navigation. Figure 11 shows the estimation ofaccelerometer bias, scale factor error, and misalignment errorin integrated navigation. Figure 12 shows the estimation ofposition error of the traditional integrated navigation modeland the proposed navigation model in the whole process,including the east position error, the north position error,and the upward position error.

From Figures 10–12, observable error estimation sta-tus of IMU in the integrated navigation process is shownin Table 6.

As can be seen from Table 6, there are 7 observable errorsof IMU of which the estimation accuracy is more than 80%.There are 4 observable errors of which the estimation accu-racy is between 50% and 80%, and there are 4 observableerrors between 20% and 50%. Consequently, most of theobservable errors of IMU are effectively estimated in theintegrated navigation process.

From the test results above, we can conclude that after500 seconds integrated navigation and 200 seconds inertialnavigation, the position errors of the east, north, and upwardof the traditional model are 798.3m, 550.6m, and 207m,respectively. The east, north, and upward position errors ofthe proposed integrated model are 592.2m, 444.8m, and176.6m, respectively. The navigation accuracy of the east,north, and upward positions of the proposed modelimproved by 25.8%, 19.2%, and 14.7%, respectively. Conse-quently, the integrated navigation method proposed in thispaper can achieve higher navigation accuracy. So the inte-grated navigation model proposed in this paper is superiorand effective.

5. Conclusions

The influence of scale factor error and misalignment error onnavigation accuracy is analyzed in this paper. Based on theanalysis, scale factor error and misalignment error are con-sidered in the error state vector. Then the observability ofscale factor error and misalignment error is analyzed com-bined with typical airborne movement. The integratedsystem is optimized according to the observability analysisresults. Finally, this method is verified by numerical simula-tion and turntable test. The results all show that the INS/GPS integrated navigation model proposed in this paper

aBxaByaBz

−100

0

100

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300

400

500

600Es

timat

ion

of ac

cele

rom

eter

bia

s (�휇

g)

100 200 300 400 5000t (s)

(a)

aSFyaSFz

0

100

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Estim

atio

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100 200 3000 500400t (s)

(b)

t (s)100 200 300 400 5000

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Estim

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acce

lero

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er m

isalig

nmen

t (”)

aMAxyaMAzx

(c)

Figure 6: Estimation of accelerometer errors. (a) Estimation ofaccelerometer bias. (b) Estimation of accelerometer scale factor.(c) Estimation of accelerometer misalignment.

10 Journal of Sensors

True valueProposed integrated model

0

5000

10000

15000

20000

25000Ea

st po

sitio

n (m

)

600 700300 400 500100 2000t (s)

(a)

True valueProposed integrated model

0

2000

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6000

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10000

12000

Nor

th p

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)

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(b)

True valueProposed integrated model

0

500

1000

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2000

2500

Upw

ard

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(m)

600 700300 400 500100 2000t (s)

(c)

Traditional integrated modelProposed integrated model

−50

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400Ea

st po

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ror (

m)

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(d)

Traditional integrated modelProposed integrated model

−250

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−100

−50

0

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ror (

m)

600 700300 400 500100 2000t (s)

(e)

Traditional integrated modeProposed integrated model

−140

−120

−100

−80

−60

−40

−20

0

20

Upw

ard

posit

ion

erro

r (m

)

600 700300 400 500100 2000t (s)

(f)

Figure 7: Estimation of position. (a) Calculation of east position. (b) Calculation of north position. (c) Calculation of upward position.(d) Estimation of east position error. (e) Estimation of north position error. (f) Estimation of upward position error.

11Journal of Sensors

gBxgBygBz

−0.4−0.2

0.00.20.40.60.81.01.21.4

Estim

atio

n of

gyr

osco

pe b

ias (

°/h)

0 200 300 400 500100t (s)

(a)

gSFxgSFy

100 200 300 4000 500t (s)

−50

0

50

100

150

200

250

300

350

Estim

atio

n of

gyr

osco

pe sc

ale f

acto

r (pp

m)

(b)

gMAxzgMAyxgMAyz

100 200 300 400 5000t (s)

−40

−20

0

20

40

60

80

Estim

atio

n of

gyr

osco

pe m

isalig

nmen

t (”)

(c)

Figure 10: Estimation of gyroscope errors. (a) Estimation ofgyroscope bias. (b) Estimation of gyroscope scale factor.(c) Estimation of gyroscope misalignment.

Table 5: Rotation parameter setting of turntable.

Time Dynamic situation Rate of rotation

0~20 Still

20~25 Pitch rotate 30° 6°/s

25~125 Still

125~130 Pitch rotate −30° −6°/s130~330 Still

330~350 Roll rotate 20° 1°/s

350~395 Yaw rotate 90° 2°/s

395~415 Roll rotate −20° −1°/s415~700 Still

Table 4: Error values of IMU.

Gyroscope errors

Bias 1°/h

Scale factor 300 ppm

Misalignment 40″Accelerometer errors

Bias 500μg

Scale factor 300 ppm

Misalignment 40″

Figure 9: Installation of the integrated system and GPS antenna.

Figure 8: Installation of the test equipment.

12 Journal of Sensors

Traditional integrated modelProposed integrated model

−800

−600

−400

−200

0

East

posit

ion

erro

r (m

)

0 200 300 400 500 600 700100t (s)

(a)

Traditional integrated modeProposed integrated model

100 200 300 400 500 600 7000t (s)

0

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Nor

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ror (

m)

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Traditional integrated modelProposed integrated model

−200

−150

−100

−50

0

Upw

ard

posit

ion

erro

r (m

)

100 200 300 400 500 600 7000t (s)

(c)

Figure 12: Estimation of position errors. (a) Estimation of theeast position error. (b) Estimation of the north position error.(c) Estimation of the upward position error.

aBxaByaBz

−400

−200

0

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600

800

1000

1200Es

timat

ion

of ac

cele

rom

eter

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s (�휇

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0 200 300 400 500100t (s)

(a)

aSFyaSFz

−500

50100150200250300350400

Estim

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acce

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m)

100 200 300 5000 400t (s)

(b)

aMAxyaMAzx

−10

−5

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5

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Estim

atio

n of

acce

lero

met

er m

isalig

nmen

t (”)

100 5000 400200 300t (s)

(c)

Figure 11: Estimation of accelerometer errors. (a) Estimation ofaccelerometer bias. (b) Estimation of accelerometer scale factor.(c) Estimation of accelerometer misalignment.

13Journal of Sensors

can obtain more effective estimation of IMU errors thantraditional integrated navigation model. In addition, whenGPS becomes invalid, the proposed integrated model canachieve higher navigation accuracy.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

References

[1] H. Han, J. Wang, and M. Du, “A fast SINS initialalignment method based on RTS forward and backwardresolution,” Journal of Sensors, vol. 2017, Article ID7161858, 11 pages, 2017.

[2] X. Wang andW. Ni, “An improved particle filter and its appli-cation to an INS/GPS integrated navigation system in a seriousnoisy scenario,”Measurement Science and Technology, vol. 27,no. 9, article 095005, 2016.

[3] G. Grunwald, M. Bakuła, A. Ciećko, and R. Kaźmierczak,“Examination of GPS/EGNOS integrity in north-easternPoland,” IET Radar, Sonar & Navigation, vol. 10, no. 1,pp. 114–121, 2016.

[4] Y. Li, G. Sun, and W. Jiang, GPS/BeiDou/INS Performance inTwo Hemispheres, Inside GNSS, NJ, USA, 2013.

[5] R. T. Kelley, N. A. Carlson, and S. Berning, “Integrated inertialnetwork,” in Proceedings of 1994 IEEE Position, Location andNavigation Symposium - PLANS'94, pp. 439–446, Las Vegas,NV, USA, April 1994.

[6] R. H. Liu and L. Z. Liang, “Status and prospect of INS/CNS/GNSS integrated navigation technology,” Modern Navigation,vol. 28, no. 6, pp. 62–65, 2014.

[7] H. J. Lei and Y. C. Zhang, Review of Airborne Inertial Naviga-tion Technology, Xi’an Flight Automatic Control ResearchInstitute, Xi’an, China, 2016.

[8] Y. Bai, Q. Sun, L. Du, M. Yu, and J. Bai, “Two laboratorymethods for the calibration of GPS speed meters,” Measure-ment Science and Technology, vol. 26, no. 1, article 015005,2015.

[9] S. Han and J. Wang, “Land vehicle navigation with the integra-tion of GPS and reduced INS: performance improvement withvelocity aiding,” The Journal of Navigation, vol. 63, no. 1,pp. 153–166, 2010.

[10] S. Han and J. Wang, “A novel initial alignment scheme forlow-cost INS aided by GPS for land vehicle applications,”The Journal of Navigation, vol. 63, no. 4, pp. 663–680, 2010.

[11] H. Peng, Z. Xiong, R. Wang, J. Liu, and J. Wang, “Dynamicparameter identification research on the installation errorsand the scale factor errors of the IMU,” Chinese Space Scienceand Technology, vol. 34, no. 1, pp. 42–49, 2014.

[12] G. Zhou, J. Wang, and K. Li, “Analysis of singular value underKalman filtering for strapdown inertial system online calibra-tion,” in Lecture Notes in Electrical Engineering, pp. 779–784,Springer, Berlin, Heidelberg, 2011.

[13] Y. Y. Qin, Inertial Navigation, Science Press, Beijing, China,2015.

[14] X. Zhang, H. Shi, and C. Zhang, “Integrated navigationmethod based on inertial navigation system and Lidar,” Opti-cal Engineering, vol. 55, no. 4, article 044102, 2016.

[15] X. L. Wang and Y. F. Li, SINS/GPS Integrated Navigation Tech-nology, Beihang University Press, Beijing, China, 2014.

[16] H. N. Wang, Principles and Applications of GPS Navigation,Science Press, Beijing, China, 2003.

[17] Q. Wang, W. Gao, M. Diao, F. Yu, and Y. Li, “Coarse align-ment of a shipborne strapdown inertial navigation systemusing star sensor,” IET Science, Measurement & Technology,vol. 9, pp. 852–820, 2016.

[18] W. Feng and J. Wang, “Design and implementation of flighttrack simulation system,” Computer Simulation, vol. 12,pp. 47–50, 2010.

[19] F. Xu, J. C. Fang, and W. Quan, “Hardware in-the-loop simu-lation of SINS/CNS/GPS integrated navigation system,” Jour-nal of System Simulation, vol. 20, no. 2, pp. 332–385, 2008.

Table 6: Observable error estimation status of IMU.

Observable error Test value Estimated valueAccuracy ofestimation

gBx 1°/h 1.1°/h 91%

gBy 1°/h 1.1°/h 91%

gBz 1°/h 0.5°/h 50%

gSFx 300 ppm 327 ppm 90%

gSFy 300 ppm 214 ppm 71%

gMAxz 40″ 51″ 78%

gMAyx 40″ 38″ 95%

gMAyz 40″ −43″ 93%

aBx 500μg 602μg 83%

aBy 500μg 702μg 71%

aBz 500μg 107μg 21%

gSFy 300 ppm 369 ppm 81%

gSFz 300 ppm 62 ppm 21%

aMAxy 40″ 13″ 33%

aMAzx 40″ −8″ 20%

14 Journal of Sensors

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