Study on GPS attitude determination system aided INS using adaptive Kalman filter

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INSTITUTE OF PHYSICS PUBLISHING MEASUREMENT SCIENCE AND TECHNOLOGY

Meas. Sci. Technol. 16 (2005) 2072–2079 doi:10.1088/0957-0233/16/10/024

Study on GPS attitude determinationsystem aided INS using adaptiveKalman filterHongwei Bian1,2, Zhihua Jin1 and Weifeng Tian1

1 Department of Information Measurement Technology and Instruments,Shanghai Jiaotong University, Shanghai 200030, People’s Republic of China2 Department of Electrical Engineering, Naval University of Engineering, Wuhan 430033,People’s Republic of China

Received 24 February 2005, in final form 1 August 2005Published 14 September 2005Online at stacks.iop.org/MST/16/2072

AbstractA marine INS/GPS (inertial navigation system/global positioning system)adaptive navigation system is presented in this paper. The GPS with twoantennae providing vessel attitude is selected as the auxiliary system to fusewith INS. The Kalman filter is the most frequently used algorithm in theintegrated navigation system, which is capable of estimating INS errorsonline based on the measured errors between INS and GPS. Theconventional Kalman filter (CKF) assumes that the statistics of the noise ofeach sensor are given. As long as the noise distributions do not change, theKalman filter will give the optimal estimation. However, the GPS receiverwill be disturbed easily and thus temporally changing measurement noisewill join into the outputs of GPS, which will lead to performancedegradation of the Kalman filter. Many researchers introduce a fuzzy logiccontrol method into innovation-based adaptive estimation Kalman filtering(IAE-AKF) algorithm, and accordingly propose various adaptive Kalmanfilters. However, how to design the fuzzy logic controller is a verycomplicated problem, which is still without a convincing solution. A novelIAE-AKF is proposed herein, which is based on the maximum likelihoodcriterion for the proper computation of the filter innovation covariance andhence of the filter gain. The approach is direct and simple without having toestablish fuzzy inference rules. After having deduced the proposedIAE-AKF algorithm theoretically in detail, the approach is tested in thedeveloped INS/GPS integrated marine navigation system. Real field testresults show that the adaptive Kalman filter outperforms the CKF withhigher accuracy and robustness. It is demonstrated that this proposedapproach is a valid solution for the unknown changing measurement noiseexisting in the Kalman filter.

Keywords: inertial navigation system, global positioning system, integratednavigation system, adaptive Kalman filter

1. Introduction

Presently, the inertial navigation system (INS) and globalpositioning system (GPS) are widely used for marinenavigation applications around the world. Considering thatboth systems possess complementary working characteristics,

booming attention is proposed, which focuses on findingeffective means to combine the two different systems into anintegrated navigation system with higher accuracy and betterperformance. The ordinary GPS receiver simply providesthe position and velocity of a vessel, which is often chosenas the external reference for INS adjustment, and is enough to

0957-0233/05/102072+08$30.00 © 2005 IOP Publishing Ltd Printed in the UK 2072

Study on GPS attitude determination system aided INS using adaptive Kalman filter

satisfy the civilian navigation [1]. In this paper, GPS providingaccurate vessel heading and pitch as well as position andvelocity is employed to be integrated with INS to constitutean outstanding marine integrated navigation system for asurveying ship, which requires higher accuracy both in positionand attitude.

The Kalman filter is the most frequently used algorithmin the integrated navigation system, which is capable ofestimating INS errors online based on the measured errorsbetween INS and GPS. The conventional Kalman filterassumes that the navigation equations and the statistics ofthe noise on each sensor are given. As long as the sensorsand noise distributions do not change, the Kalman filter willgive the optimal estimation [2]. GPS has superior long-termperformance but poor short-term accuracy. It is easy for theGPS signal to be disturbed because of multi-path effect, worsePDOP (position dilution of precision) value, etc. Differentfrom the former GPS based on pseudo-range measurement,the GPS attitude determining system offers more accuratereal-time azimuth using the measurement of GPS carrier phasesignals based on real-time kinematics/on-the-fly (RTK/OTF)solutions. The attitude accuracy of the system is closely relatedto the calculation of distance between two antennae [3]. Thus,the attitude outputs of GPS are more prone to be disturbed bythe temporally changing signal noise than the position output,which will lead to significant performance degradation of theKalman filter.

In order to improve the practicability of the Kalman filter,different adaptive schemes have been proposed. In the pastfew years, adaptive Kalman filtering based on innovation-based adaptive estimation (IAE) has become one of themajor approaches under study. In the IAE approach [4],the noise distribution matrices employed in the Kalman filterare adjusted as measurements change with time. The maingoal of this approach (or these approaches) is to design avalid algorithm to adjust the noise matrices approximately.Many researchers introduce fuzzy logic control (FLC) intothe IAE algorithm, and accordingly propose various adaptiveKalman filtering methods. Sasiadek employed FLC to adjustthe exponential weights in a weighted Kalman filter basedon the change of innovation mean and covariance [5]. Toacquire better system stability, Loebis used FLC to measuremore accurate covariance R by diagnosing the innovationcovariance drift and calculating the appropriate increment[6]. Zhang used a similar method to adjust the gain indexof measured noise [7]. The approaches mentioned abovedecrease the computation time of the algorithm remarkablywithout increasing the system state dimension. Simulationsverified their good robustness and accuracy. However, thekey point, that is how to establish the fuzzy inference rulesand select the membership function, is not provided with aconvincing solution.

Based on the conventional Kalman filter, a real-time innovation-based adaptive estimation Kalman filter isproposed in this paper to improve the system’s overallperformance when GPS measurement noise changes abruptly.The novel approach adjusts the Kalman gain directly fromthe computation of real-time innovation without establishingthe fuzzy inference rules and provides higher accuracy,stability and additionally less computation time. Based on

the theoretical analysis and real field INS/GPS data, theeffectiveness of the proposed method is verified.

In this paper, this section introduced related researchingbackgrounds. Section 2 deduces the proposed IAEKalman filter algorithm in detail. Section 3 outlines theintegrated navigation system based on INS and GPS attitudedetermination system, and gives the mathematical model forthe integrated navigation system. The test results and analysisare presented in section 4. Finally, the conclusion is given insection 5.

2. Adaptive Kalman filtering algorithm

2.1. Innovation-based adaptive estimation Kalman filteringalgorithm

Based on a group of recursive formulae, the Kalmanfilter was proposed in 1960 for the purpose of optimallyestimating the system’s true states among practical noise.The Kalman gain matrix K, the key to determining theadjustability of the Kalman filter, is unrelated to the real-time system measurements but only depends on the priorlyknown transmitted matrix �, measurement matrix H, processnoise covariance matrix R and measurement noise covariancematrix Q. When �, H, R and Q are mismatched or varywith time, the optimal estimation condition will no longerbe satisfied, the fault computation of K and degradationof the performance of the filter will happen accordingly.To rehabilitate the adjustability of K, in this section, theinnovation-based adaptive estimation algorithm is proposed.

The innovation is the difference between the realmeasurement received by the filter and its predicted value. Theinnovation of the Kalman filter is a theoretical zero-mean whitenoise sequence if it reaches the stable state ideally [8]. In thecase that the filter is abnormal, the innovation statistic characterbecomes complicated. Hence, the innovation can be used asa criterion to detect the divergence of the filter and implementthe adjustment of the system. It is a significant approachto accomplish an adaptive Kalman filter by introducing theinnovation sequence to adjust the calculation of the Kalmangain. The novel adaptive Kalman filter proposed herein isto calculate the Kalman gain K directly from the statisticcovariance of the real-time residual online.

For a discrete linear system, the state equation andmeasurement equation are

Xk = �k,k−1Xk−1 + Wk−1,

Zk = HkXk + Vk,

where �k,k−1 is the system state transition matrix, Hk is themeasurement matrix, Wk is the process noise vector and Vk isthe measurement noise vector. Both Vk and Wk are assumedto be uncorrelated zero-mean Gaussian white noise sequenceswith covariance given by Cov[Wk, Vj ] = 0, Cov[Wk,Wj ] =Qkδkj , Cov[Vk, Vj ] = Rkδkj and

δkj ={

1, k = j,

0, k �= j.

The recursive formulae of an innovation-based conventionalKalman filter are as follows:

Xk+1 = �Xk, (1)

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H Bian et al

P −k+1 = �P +

k�T + Qk, (2)

Kk = P −k HT

k C−1vk , (3)

Xk = X−k + Kkvk, (4)

P +k = [1 − KkHk]P −

k . (5)

In equations (1)–(5), vk denotes the innovation. Theexpression of innovation vk and covariance of innovation Cvk

can be calculated by equations (6) and (7).

vk = Zk − HkX−k , (6)

Cvk = HkP−k HT

k + Rk. (7)

Three fundamental approximate assumptions about the IAE-AKF are listed as follows:

1. Qk is stabilized and Vk corresponds to piecewise constantnon-zero white noise.

2. The changing intensity of covariance of Vk is an unknownconstant over each sampling period and is independentfrom period to period.

3. The innovation sequence is ergodic inside the movingestimation windows with size N.

Let Cvk be the optimal estimation of Cvk in the slidingsampling windows with length N. We prove herein that Cvk

can be calculated through the expression below:

Cvk = 1

N

k∑j=j0

vjvtj . (8)

By substituting equation (8) into (3), we obtain the modifiedproposed IAE adaptive Kalman gain K∗

k :

K∗k = P −

k HTk C−1

vk . (9)

Obviously, the approach is direct and simple without having toestablish fuzzy inference rules. In the subsequent subsection,the theoretical proof for equation (8) is given in detail.

2.2. Maximum likelihood estimator of innovation adaptiveKalman filter

The maximum likelihood (ML) concept [9] is used in thissubsection to derive the IAE adaptive Kalman filter. Letr = {r11, r22, . . . , rnn}, where rii is the diagonal elementof measurement noise R. According to the central limittheorem, the probability density function of the measurementsconditioned on the adaptive parameter r at the specific epochk is a Gaussian distribution:

p(Z|r)k = 1√(2π)m|Cvk|

e− 12 vT

k C−1vk vk , (10)

where m is the number of measurements, | · | is the determinantoperator and e is the base of natural logarithms. Thelogarithmic form of (10) is

ln p(Z|r)k = − 12

{m × ln(2π) + ln(|Cvk|) + vT

k C−1vk vk

}. (11)

After multiplying by –2, summation and neglecting theconstant term, the ML condition becomes

k∑j=j0

ln |Cvj | +k∑

j=j0

vTj C−1

vj vj = min . (12)

The above equation describes the best estimation as the one thathas the maximum likelihood based on the adaptive r. Matrix

differential calculus equates its element r to zero. From thematrix differential calculus formula ∂p

∂r= 0, considering

∂ ln|Cvj |∂r

= 1

|Cvj |∂|Cvj |

∂r= tr

{C−1

vj

∂Cvj

∂r

}and

∂ ln C−1vj

∂r= −C−1

vj

∂Cvj

∂rC−1

vj ,

where tr is the matrix trace operator, equation (13) can bederived as follows:

k∑j=j0

[tr

{C−1

vj

∂Cvj

∂r

}− vT

j C−1vj

∂Cvj

∂rC−1

vj vj

]= 0. (13)

The partial derivation of equation (7) with respect to r yields

∂Cvk

∂rk

= Hk

∂P −k

∂rk

HTk +

∂Rk

∂rk

. (14)

It is also known that P −k = �Pk�

T + Qk−1. Assuming thatthe process inside the estimation windows is in steady stateand Q is completely known and independent of r, the firstterm of equation (14) can be neglected. Considering thatthe adaptive parameters r are the variances of the updatemeasurements inside the estimation windows, the secondterm of equation (14) can be equal to unit matrix I. Thus,equation (14) reduces to

∂Cvk

∂rk

= I. (15)

Now, substitute equation (15) into equation (13) and expandit. The resulting expression, equation (16), is the ML equationfor the adaptive Kalman filter:

k∑j=j0

[tr{C−1

vj

} − vTj C−1

vj C−1vj vj

]

=k∑

j=j0

tr{C−1

vj

[Cvj − vjv

Tj

]C−1

vj

} = 0. (16)

From the above formula and under the assumption of anergodic innovation sequence inside the estimation window,the expression for the estimated V–C matrix of the innovationsequence as in equation (8) can be obtained.

Cvk = 1

N

k∑j=j0

vjvtj .

3. INS/GPS integrated navigation system

3.1. Introduction of INS/GPS integrated navigation system

An accurate INS/GPS integrated marine navigation system iscurrently under development by Shanghai Jiaotong University.The gimbaled INS, utilized on the survey ship, consists ofthree high-precision single-degree float-type gyroscopes andthree quartz accelerometers and is capable of providing trueheading and attitude better than 0.05◦ (RMS). A TrimbleMS860 DGPS receiver, offering accurate yaw and pitch as wellas position and velocity of a ship, is selected as the auxiliarysystem for INS. RTK positioning utilized in MS860 is basedon at least two GPS receivers—a base receiver and one ormore rover receivers. The base receiver takes measurements

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(a) (b)

Figure 1. Trimble GPS attitude determination system (MS860): (a) static performance testing and (b) dynamic performance testing on thesea.

from satellites in view and then broadcasts them, togetherwith its location, to the rover receiver. The rover receiveralso collects measurements from the satellites in view andprocesses them with the base station data. Figure 1(a) showsthe static performance testing of the GPS with a fixed baselineon the rigid bracket on the ground. Figure 1(b) shows thedynamic performance trial on the South China Sea in whichthe GPS was installed on a ship. Performance analysisfrom different tests shows that the GPS attitude determiningsystem with 17 m antenna separation is competent for offeringreal-time azimuth determination within 0.03◦ (1σ ) and pitchdetermination within 0.05◦ (1σ ), respectively. Based onthe gimbaled INS and Trimble MS860 DGPS, the marineINS/GPS integrated navigation system is built.

3.2. The model of INS/GPS integrated system

3.2.1. Error state equation. The gimbaled INS employsthe east–north–up (ENU) coordinate system and the kineticscharacteristic of GPS employs the first-order Markov processequation. The state equation of the integrated system can bewritten as follows:

X =[X INS

XGPS

]= AX + BW =

[FINS 0

0 FGPS

][X INS

XGPS

]+ BW ,

(17)

where X INS = [δLI δλI δVEI δVNI φe φn φu ∇e ∇n εe εn εu]T ,and δLI and δλI represent INS latitude error and longitudeerror, respectively; δVEI and δVNI are east and north velocityerrors, respectively; φe, φn and φu are east, north and azimuthmisalignment angles, respectively; ∇e and ∇n are the east andnorth accelerator biases, respectively; εe, εn and εu representthe constant drift rates of east, north and azimuth gyros,respectively. XGPS = [δLG δλG δVEG δVNG δψG δθG]T , andδLG and δλG represent GPS latitude error and longitude error,respectively; δVEG and δVNG are east and north velocity errors,respectively; δψG and δθG are heading error and pitch error,respectively. FINS is a 12 × 12 matrix, see equation (18) [10].

FINS =

0 0 0 1RM

Ve sec(L) tan(L)

RN0 sec(L)

RN0

2ωieVn cos(L) + VeVn sec2 (L)

RN0 Vn tan(L)

RN2ωie sin(L) + Ve tan(L)

RN

−2ωieVe cos(L) − V 2e sec2 (L)

RN0 −2ωie sin(L) − Ve tan(L)

RN0

0 0 0 − 1RM

−ωie sin(L) 0 1RN

0

−ωie cos(L) + Ve sec2 (L)

RN0 tan(L)

RN0

0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 −g 0 1 0 0 0 0g 0 0 0 1 0 0 00 ωie sin(L) + Ve tan(L)

RN−ωie cos(L) − Ve

RN0 0 1 0 0

−ωie sin(L) − Ve tan(L)

RN0 − Vn

RN0 0 0 1 0

ωie cos(L) + VeRN

VnRM

0 0 0 0 0 1

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

,

(18)

where g is the local gravity, L is the local latitude, ωie

is the earth rate, Ve and Vn are east and north velocities,respectively, and R is the radius of the earth. BINS =[02×12; 05×7I5×5; 05×12]T . The process noises are zero-mean Gaussian processes with covariance QINS. WINS =[01×7 Wδ∇e Wδ∇n Wεe Wεn Wεu

]T. FGPS is a 7 × 7

matrix, see equation (19).

FGPS = diag

[− 1

τLG

− 1

τλG

− 1

τVEG

− 1

τVNG

− 1

τψG

− 1

τθG

],

(19)

where τ with different subscripts represents differenttime correlated constants corresponding to the GPS errorstates.

3.2.2. Measurement equation. Let ψI = ψ + δψI, θI = θ +δθI and γI = γ + δγI represent the INS measurements of ship’syaw, pitch and roll, respectively. Let ψG = ψ + δψG and θG =θ + δθG represent the GPS measurements of ship’s yaw andpitch, respectively. We assume that the measurement noiseVGPS is a zero-mean Gaussian process under the conditionthat covariance RGPS and VGPS,WINS are independent. Themeasurement equation can be written as follows:

Z =

LI − LG

λI − λG

vEI − vEG

vNI − vNG

ψI − ψG

θI − θG

=

δLI − δLG

δλI − δλG

δvEI − δvEG

δvNI − δvNG

δψI − δψG

δθI − δθG

= H

[X INS

XGPS

]+ V .

(20)

Let the coordinate system of the gimbaled INS inertialstabilized platform be represented by OXpYpZp, the shipcoordinate system be represented by OXbYbZb and thegeographic coordinate system be represented by OXtYtZt.Let the ship’s heading, pitch and roll be represented byψ , θ and γ , respectively. To determine the misalignment

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Figure 2. INS and GPS positions: (a) INS position data; (b) GPS position data.

angle of the inertial stabilized platform is to determine therelationship between OXpYpZp and OXtYtZt. From thecoordinate transform theorem, C

p

b = Cpt × Ct

b, namely

Ctb =

[C11 C12 C13

C21 C22 C23

C31 C32 C33

]

=[

cos γ cos ψ − sin γ sin θ sin ψ − cos θ sin ψ sin γ cos ψ + cos γ sin θ sin ψ

cos γ sin ψ + sin γ sin θ cos ψ cos θ cos ψ sin γ sin ψ − cos γ sin θ cos ψ

− sin γ cos θ sin θ cos γ cos θ

].

(21)

And H can be written as follows [9]:

H =

1 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 00 0 1 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 00 0 0 1 0 0 0 0 0 0 0 0 0 0 0 −1 0 0

0 0 0 0 C12C32

C212+C2

22

C22C32

C212+C2

22−1 0 0 0 0 0 0 0 0 0 −1 0

0 0 0 0 − C22√1−C2

32

C12√1−C2

32

0 0 0 0 0 0 0 0 0 0 0 −1

.

(22)

4. Test and results

4.1. Introduction of the test

To verify the better performance of the developed IAE adaptiveKalman filter than the conventional Kalman filter, tests ofthe INS/GPS integrated navigation system are carried outon a surveying ship. The tests are divided into two phases,static test and dynamic test. Before static test, the gimbaledINS and MS860 DGPS receiver were well installed in thecabin of the ship. With the aid of a highly accurate opticaltheodolite, GPS antennae were installed on two marks alongthe midline of the ship precisely, with an accuracy better than5 arc-seconds. The baseline of the GPS antennae is 17.04 mand the orientation of the baseline is 364.52◦ with an accuracybetter than 2 arc-seconds, which were measured by the opticalequipment relative to the North Star at night. The orientationof the baseline hence can be used as the reference of the shipto verify the final true heading accuracy of INS, GPS and theirintegration.

The static test was accomplished when the surveying shipwas anchored on the slipway in Shanghai boatyard, and thedynamic test was carried out when the ship was voyagingaround Zhoushan Islands on the East China Sea with thecalibrated GPS and INS. During both tests, the output ofINS and GPS were collected through an online computer.INS outputs latitude, longitude, velocity and three attitudes

Table 1. The statistical data of the noise covariance of the GPSheading during the test.

Time range (s) 0–800 800–1600 1600–2400 2400–3600

Heading error (deg) 0.0146 0.0314 0.0342 0.0155Baseline error (m) 0.0065 0.0096 0.0118 0.0098

of the ship in 20 Hz. GPS outputs latitude, longitude, headingand pitch of the ship in 1 Hz. Both of the systems employGPS UTC to accomplish data time synchronization with anaccuracy better than 1 µs.

4.2. Static test and analysis

The static test was carried out after both the INS and GPShad been well installed on the ship and their installation biaseshad been correctly compensated. Figures 2 and 3 show onegroup of the sampled latitude, longitude, heading and pitchof INS and GPS in the static test, respectively. From thoseplots, we can see that as time varies, the position errors ofINS will diverge gradually. The latitude errors of INS followthe well-known Schuler period with some small measurementnoise. In contrast, the position errors of GPS are stable in thelong term, with an accuracy of 10.94 m (1σ ). As for the GPSheading and pitch accuracies, it can maintain accuracies of0.028◦ (1σ ) and 0.052◦ (1σ ), respectively. However, even inthis situation, the covariance of the GPS noise is time varyingbecause of the changing numbers of satellite in view or multi-path effect, etc. The value of position dilution of precision(PDOP) of GPS can be used as a criterion to evaluate GPScurrent noise. Figure 3(e) shows the GPS PDOP value of thetest. When the PDOP value is beyond 3 (for example, at timesbetween 800 s and 1100 s), the quality of the GPS signal willbecome worse. Figure 3(f ) shows the corresponding baselineoutput of GPS. Since the accuracy of baseline calculation hasa close relationship with the accuracy of GPS attitude output,if the baseline becomes unstable (at times between 800 s and2500 s), obvious errors will join into GPS heading and pitch.Table 1 gives the statistical data about the noise covariance ofthe GPS heading during the test.

The IAE Kalman filter and conventional Kalman filterare designed for the GPS/INS integrated navigation system,respectively, to verify the proposed approach’s effectiveness.From the plots in figure 4, the CKF is extremely sensitive toexternal disturbance. The estimation capability of the CKFdegrades obviously once the GPS signal becomes unstable

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Study on GPS attitude determination system aided INS using adaptive Kalman filter

Figure 3. INS and GPS attitude data: (a) INS heading; (b) INS pitch; (c) GPS heading; (d) GPS pitch; (e) GPS PDOP; (f ) GPS baseline.

Figure 4. Performance comparison between CKF (dotted) and IAE-AKF (solid) in static test: (a) latitude estimation error; (b) longitudeestimation error; (c) heading estimation error; (d) pitch estimation error.

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Figure 5. GPS and INS data and the performance comparison between CKF and IAE-AKF in the dynamic test: (a) INS and GPS position;(b) position and the estimation comparison; (c) INS and GPS heading; (d) heading and the estimation comparison; (e) INS and GPS pitch;(f ) pitch and the estimation comparison; (g) comparison of estimated heading error; (h) comparison of estimated pitch error.

(for example, at time 1000 s). Especially between 800 sand 2400 s, rough fluctuations happen in the estimation ofINS heading and pitch errors (the estimation of heading andpitch even beyond 1.2◦ and 0.4◦, respectively). Consideringthe INS and GPS practical accuracy, the CKF has actuallyalready lost its effect. Under the same situation, however,the performance of the proposed IAE adaptive Kalman filterremains stable. From the solid curves of figure 4, we cansee that the AKF estimations vary gradually and no obviousfluctuations happen. The filter shows good robustness to dealwith the changing measurements of GPS. The INS errors are

found obtaining good estimation. Analysis of other groups ofstatic test data gives similar results. The overall statistical dataon the performance of the integrated navigation system usingCKF and AKF are listed in table 2.

4.3. Dynamic test and analysis

The dynamic test was carried out while the ship was voyagingaround Zhoushan Islands on the East China Sea. There is oneimportant thing that should be emphasized herein. To evaluatethe overall accuracy of different algorithms employed in the

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Study on GPS attitude determination system aided INS using adaptive Kalman filter

Table 2. Statistical data of accuracy comparison between IAE-AKFand CKF for the estimation of INS error.

Accuracy of the filter estimation for INS error (STD)

Algorithm Latitude (m) Longitude (m) Heading (deg) Pitch (deg)

IAE-AKF 3.05 2.88 0.015 0.011CKF 36.77 34.36 0.24 0.15

Table 3. Statistical data for the comparison of the estimation of INSerrors between IAE-AKF and CKF.

Filter estimation for INS error (STD)

Algorithm Latitude (m) Longitude (m) Heading (deg) Pitch (deg)

IAE-AKF 386.9 2961.0 0.036 0.028CKF 457.1 3043.3 0.43 0.41

integrated system, true orientation and position of the shipmust be indispensable. In reality, however, only in the staticenvironment can the true reference of the ship be measuredthrough external highly accurate measurements. Since the INSand GPS employed in this system possess high accuracy, a testreference with higher accuracy on the ship at sea becomesunreachable. Thus, in the dynamic environment, the realkinetic states of ship still remain unknown. However, inthis situation, the GPS and INS data of the dynamic testare still helpful to verify the effectiveness of the proposedintegrated algorithm in the navigation system. Figure 5 showsa group of dynamic test data. The comparison of INS andGPS positions is plotted in figure 5(a), and the comparisonsof INS and GPS heading and pitch are plotted in figures 5(c)and (e), respectively. The comparative results of the integratednavigation system based on CKF and IAE-AKF are plotted infigures 5(b), (d) and (f ).

From figure 5(b), we can see that the INS position bothin CKF and IAE-AKF was revised and approached the GPSdata. The revised position of IAE-AKF is closer to GPSdata than that of the CKF. Considering the GPS has higherposition accuracy, the position of IAE-AKF can be regardedas obtaining higher accuracy than that of the CKF. Figure 5(d)

shows the comparison between the INS heading and itsestimation based on IAE-AKF and CKF. To make it clear,figure 5(g) gives the heading difference between INS and GPSand the estimation of INS heading error by different methods.We can see that the estimations of the CKF are vibrated and aregreater than the heading difference. When the ship is turning,the estimations of the CKF become incredibly larger. The CKFis unable to fit in with the unknown changing measurementnoise and becomes untrustworthy. Similar results appear whenCKF tries to estimate the error of INS pitch, see figure 5(h).The performance of the CKF is poor. In contrast, the headingand pitch errors estimated by IAE-AKF vary in a small rangeand show good stability. Even when the heading of the shipchanges rapidly, the estimation can maintain stability and itsvalue remains below 0.05◦. Similar results are obtained byanalysing other groups of dynamic test data.

Table 3 shows the overall statistical analysis for the outputof the integrated navigation system based on CKF and IAE-AKF, respectively. The estimations of INS position error ofboth methods are about 3000 m. The result is acceptablebecause there are large errors in the positions of INS. When

IAE-AKF is employed in the integrated system, the estimatederror of INS heading is 0.036◦ and that of INS pitch is 0.028◦.Considering that the attitude accuracy of INS is better than0.05◦, this result is reasonable and acceptable. However, itis unacceptable that the estimation of INS heading error is0.43◦ and that of pitch error is 0.41◦ when employing CKF.Compared to the result of CKF, from the comparison of curvesand tables, we can see that the proposed IAE-AKF algorithmhas a more accurate estimation to enhance the performance ofthe integrated navigation system.

5. Conclusions

This paper presents an adaptive INS/GPS marine navigationsystem. The GPS with two antennae providing the attitudesof a vessel is selected as the auxiliary system to integrate withINS to obtain better estimation accuracy of INS errors. Anovel innovation-based adaptive estimation Kalman filteringapproach is proposed to solve the degradation performancecaused by GPS unstable measurement disturbances in thehybrid system. The developed adaptive Kalman filter isbased on the maximum likelihood criterion for the propercomputation of the filter innovation covariance and hence thefilter gains. Compared with the conventional Kalman filter,real test data verify the validity of the IAE-AKF algorithm.It is demonstrated that this proposed approach is capable ofproviding the developed integrated navigation system withhigher accuracy and stability.

Acknowledgments

This research is grateful for the support of the National NaturalScience Foundation Project (no 40125013 and no 40376011).

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