Research ArticleA SINS/BDS Integrated Navigation MethodBased on Classified Weighted Adaptive Filtering

Xuchao Kang ,1 Guangjun He ,2 and Xingge Li 1

1Graduate College, Air Force Engineering University, Xi’an, China2Air and Missile Defense College, Air Force Engineering University, Xi’an, China

Correspondence should be addressed to Xuchao Kang; 919454628@qq.com

Received 6 April 2019; Accepted 3 June 2019; Published 7 July 2019

Academic Editor: Carlo Renno

Copyright © 2019 Xuchao Kang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Aiming at the problem that the accuracy and stability of SINS/BDS integrated navigation system decrease due to uncertain modeland observation anomalies, a SINS/BDS integrated navigation method based on classified weighted adaptive filtering is proposed.Firstly, the innovation covariance matching technology is used to detect whether there is any abnormality in the system as awhole. Then the types of anomalies are distinguished by hypothesis test. Different types of anomalies have different effects on stateestimation. Based on the dynamic changes of innovation, different adaptive weighting methods are adopted to correct navigationinformation. The simulation results show that this method can effectively improve the fault-tolerant performance of integratednavigation system in complex environment with unknown anomaly types.When bothmodel anomalies and observation anomaliesexist, the speed and position accuracy are increased by 42% and 24% compared with the standard KF, 38% and 22% compared withthe innovation orthogonal adaptive filtering, which has higher navigation accuracy.

1. Introduction

The development of high-speed cruise aircraft puts higherdemands on the accuracy and fault tolerance of navigationsystems [1]. SINS/BDS integrated navigation system com-bines the advantages of each sub-navigation system throughfiltering algorithm. It overcomes the defects that the outputinformation errors of the SINS accumulate over time and themeasurements of BDS are easily affected by the environment.Therefore, SINS/BDS has been widely used in the designof aircraft navigation systems [2–4]. However, with thedevelopment of hypersonic vehicles, the rapid increase offlight speed makes the flight environment of the aircraftchange rapidly in a short time; it will cause the aircraft to bedisturbed by more uncertain factors. These uncertainties willmake the navigation system anomalies occur and navigationerrors increase or even diverge.

The online matching technology of innovation and itscovariance matrix is widely used in the anomaly detectionof navigation systems. Many literatures have carried out in-depth research and improvement [5–8]. In [5], an adaptive fil-tering algorithm based on the orthogonality of innovation is

proposed. According to the orthogonality of innovation, theoutliers are identified and weighted to weaken the influenceof outliers on filtering. But it uses the same weighting factorfor different outlier signals; the overall filtering accuracyis not high. In [6], the anomalous signals are classified bycomparing the theoretical and actual values of the innovationcovariance. Then different weighting matrices are used tocorrect the filter deviation caused by the outliers. It improvesthe accuracy of the navigation system, but does not considerthe influence of model error caused by uncertain factorson the filtering results. In reference [7], a filtering methodwith adaptive factor is proposed. It is considered that thecovariance matrix of residual sequence is mainly affected bythe covariance matrix of system noise. The adaptive factor isconstructed to adjust the system noise matrix in real timeto reduce the influence of uncertain dynamic model errorson navigation filtering solution. However, the influence ofmeasurement anomalies on residual covariance matrix isnot considered in this paper. When measurement anomaliesoccur, filtering errors will tend to increase.

The innovation variance can be used to test whetherthe system is abnormal. However, due to the uncertainty of

HindawiMathematical Problems in EngineeringVolume 2019, Article ID 2158351, 6 pageshttps://doi.org/10.1155/2019/2158351

2 Mathematical Problems in Engineering

interference factors, it is impossible to determine whether theanomaly that occurred is model anomaly or measurementanomaly. For the above problem, the previous references didnot do too much analysis, but designed the fault-tolerantfiltering algorithm by simply considering the innovationcovariance matching error as a known type of anomaly.However, in the case where the type of the anomaly isuncertain, this method will seriously affect the accuracy ofthe navigation information and even produce the oppositefault-tolerant effect. So a SINS/BDS integrated navigationmethod based on classification weighted adaptive filtering isdesigned in this paper. Firstly, according to the orthogonalityof the innovation, the method uses the innovation covari-ance matching technology to identify whether the system isabnormal. Based on the innovation covariance, the weightingfactors are constructed. Then the type of the anomaly is fur-ther tested. According to the test results, different weightingmethods are adopted to correct the filtering error caused bythe uncertain factors. In addition, the vector-form weightingfactors are used instead of the scalar-form weighting factorsin the weighting process to further improve the accuracy ofthe navigation filtering.

2. SINS/BDS Integrated NavigationMathematical Model

2.1. Integrated Navigation System State Equation. The ENUgeography coordinate system is selected as the basic coordi-nate system of the navigation solution, and the state equationis established as formula (1)

𝑋 (𝑘 + 1) = 𝐹 (𝑘)𝑋 (𝑘) + 𝐺 (𝑘)𝑊 (𝑘) (1)

System state variable is expressed as

𝑋 = [𝛿𝐿, 𝛿𝜆, 𝛿ℎ, 𝛿𝑉𝑒, 𝛿𝑉𝑛, 𝛿𝑉𝑢, 𝜙𝑒, 𝜙𝑛, 𝜙𝑢, 𝜀𝑥, 𝜀𝑦, 𝜀𝑧, ∇𝑥, ∇𝑦,∇𝑧]T

(2)

where 𝛿𝐿, 𝛿𝜆, 𝛿ℎ represent latitude error, longitude errorand height error respectively; 𝛿𝑉𝑒, 𝛿𝑉𝑛, 𝛿𝑉𝑢 represent thevelocity error in the east, north and up directions respec-tively; 𝜙𝑒, 𝜙𝑛, 𝜙𝑢 are the mathematical platform angular error.𝜀𝑥, 𝜀𝑦, 𝜀𝑧 are the gyro random constant drift. ∇𝑥, ∇𝑦, ∇𝑧 arethe accelerometer random constant value deviation. 𝑊 =[𝑤𝜀𝑥 , 𝑤𝜀𝑦 , 𝑤𝜀𝑧 , 𝑤∇𝑥 , 𝑤∇𝑦 , 𝑤∇𝑧]T represents integrated naviga-tion system noise.

2.2. Integrated Navigation System Measurement Equation.Take the difference between the position and velocity infor-mation of the INS and the BDS as the observation mea-surement, and establish the systemmeasurement equation asformula (3)

𝑍 (𝑘) = [𝐻𝑝𝐻V]𝑋 (𝑘) + [𝑉𝑝 (𝑘)

𝑉V (𝑘)] (3)

𝐻𝑝 and 𝐻V are measurement matrices of measurementequation, which are expressed as 𝐻𝑝 = [diag(𝑅𝑀, 𝑅𝑁 cos 𝐿,1), 03×6, 03×6], 𝐻V = [03×3, diag(1, 1, 1), 03×9)]. 𝑉𝑝 and 𝑉V are

the position and velocity measurement noise of BDS in theENU geography coordinates system, which can be expressedas 𝑉𝑝 = [𝑁𝐺𝑒 , 𝑁𝐺𝑛 , 𝑁𝐺𝑢]T, 𝑉V = [V𝐺𝑒 , V𝐺𝑛 , V𝐺𝑢]T.3. Classification WeightedAdaptive Filtering Algorithm

3.1. Problem Analysis. Innovation ek can be described asek = 𝑍𝑘 − 𝐻�̂�𝑘.𝑘−1; it indicates the deviation between thereal measurement and prediction of navigation system. Theabnormality of navigation system will cause the change ofinnovation statistical characteristics [9]. According to theorthogonality theory of innovation, it can be deduced that thetheoretical innovation covariance matrix can be expressed asequation (4).

𝐿𝑘 = 𝐸 (𝑒𝑘𝑒𝑇𝑘 ) = 𝐸 ((𝑍𝑘 −𝐻�̂�𝑘,𝑘−1)⋅ (𝑍𝑘 −𝐻�̂�𝑘,𝑘−1)T) = 𝐸 ((𝐻𝑋𝑘 +𝑉𝑘 −𝐻�̂�𝑘,𝑘−1)⋅ (𝐻𝑋𝑘 +𝑉𝑘 −𝐻�̂�𝑘,𝑘−1)T) = 𝐸 ((𝐻�̃�𝑘 + 𝑉𝑘)⋅ (𝐻�̃�𝑘 +𝑉𝑘)T) = 𝐻𝑃𝑘,𝑘−1𝐻𝑇 + 𝑅𝑘

(4)

where X̃k is the system one-step prediction error and Vk isthe measurement error. The innovation covariance matchingalgorithm compares the theoretical and actual values ofthe innovation covariance and judges the system to beabnormal when the actual value deviates significantly fromthe theoretical value [9].

When the uncertainty causes the navigation system tomeasure abnormally, the measurement equation can beexpressed as formula (5).

𝑍∗𝑘 = 𝐻𝑋𝑘 +𝑉𝑘 + �𝑉𝑘 (5)

where 𝑍∗𝑘 denotes abnormal measurement; 𝑉𝑘 denotes fixedmeasurement noise, and �𝑉𝑘 denotes uncertain measure-ment noise that causes abnormal measurement. Innovationcovariance is expressed as equation (6).

𝐸 (𝑒𝑘𝑒𝑇𝑘 ) = (𝐻�̃�𝑘 +𝑉𝑘 + �𝑉𝑘) (𝐻�̃�𝑘 +𝑉𝑘 + �𝑉𝑘)T= 𝐻𝑃𝑘,𝑘−1𝐻𝑇 + 𝑅𝑘 + �𝑅𝑘

(6)

When the uncertainty causes the system model of thenavigation system to be abnormal, the equation of state canbe expressed as equation (7)

𝑋∗𝑘 = (𝐹 + �𝐹)𝑋𝑘−1 + 𝐺𝑘𝑊𝑘 (7)

State one-step prediction equation is expressed as equation(8)

�̂�∗

𝑘,𝑘−1 = (𝜑 + �𝜑)𝑋𝑘−1 = �̂�𝑘−1 + ��̂�𝑘−1 (8)

�̂�∗

𝑘,𝑘−1 represents the one-step prediction of state variablewhen the model has an abnormality, �𝜑 represents the

Mathematical Problems in Engineering 3

model error of the abnormal system. Innovation covarianceis expressed as formula (9)

𝐸 (𝑒𝑘𝑒T𝑘 ) = 𝐸 ((𝐻 (𝑋𝑘 − �̂�𝑘,𝑘−1 − ��̂�𝑘,𝑘−1) +𝑉𝑘)⋅ (𝐻 (𝑋𝑘 − �̂�𝑘,𝑘−1 − ��̂�𝑘,𝑘−1) +𝑉𝑘)T)= 𝐻𝑃𝑘,𝑘−1𝐻T +𝐻�𝑄𝑘𝐻T + 𝑅𝑘

(9)

Based on the above analysis, the actual innovationcovariance will deviate from the theoretical value regardlessof whether the model is abnormal or the measurement isabnormal. The covariance matching method can be used todetermine whether the system has an abnormality, but thetype of the anomaly cannot be determined. In theory, whenthe measurement is abnormal, the gain is usually reducedto weaken the influence of the abnormal measurement onthe filtering. When the model is abnormal, the gain isusually increased to enhance the correction effect of themeasurement on the model error [10]. Therefore, in thecase where the type of anomaly cannot be determined, thetraditional single weighted filtering methodmay produce theopposite fault-tolerant effect.

3.2. ClassifiedWeighted Adaptive Filtering Algorithm. First ofall, the one-step prediction error equation can be describedas follows.

𝑃𝑘,𝑘−1 = 𝜑𝑘,𝑘−1𝑃𝑘−1𝜑T𝑘,𝑘−1 + Γ𝑘−1𝑄𝑘−1ΓT𝑘−1 (10)

Fromequation (10), it can be seen thatwhen the systemmodelis abnormal or the statistical characteristics of the systemnoise change, equation (11) can be obtained and the one-stepprediction variance will change.

𝑃𝑘,𝑘−1 = (𝜑𝑘,𝑘−1 + �𝜑𝑘,𝑘−1)𝑃𝑘−1 (𝜑𝑘,𝑘−1 + �𝜑𝑘,𝑘−1)T+ Γ𝑘−1 (𝑄𝑘−1 + �𝑄𝑘−1) ΓT𝑘−1

(11)

Therefore, this paper simplifies the correction of the nav-igation system anomaly caused by the uncertainty modelerror into the adaptive adjustment of the one-step predictionvariance matrix. Through the adaptive weighting of themeasurement noise, the fault tolerance of the navigationsystem is measured. Filter gain and state estimation equationcan be obtained as follows.

𝐾𝑘 = 𝛽𝑘𝑃𝑘,𝑘−1𝐻T𝑘 (𝐻𝑘𝛽𝑘𝑃𝑘,𝑘−1𝐻T

𝑘 + 𝛼𝑘𝑅𝑘)−1 (12)

�̂�𝑘 = �̂�𝑘,𝑘−1 +𝐾𝑘𝑒𝑘 (13)

For the calculation of the weighting factors 𝛼 and 𝛽, differentcalculation methods will be adopted for different types ofabnormalities after judging that there is an abnormality in thewhole system.

Firstly, the innovation covariance matching method isused to judge whether the system as a whole is abnormal.

For the calculation of innovation covariance, a calculationmethod with sliding window can be used [11].

�̂�𝑘 ={{{{{{{{{

𝑘 − 1𝑘 �̂�𝑘−1 +

1𝑘𝑒𝑘𝑒T𝑘 𝑘 ≤ 𝑁

1𝑁𝑘∑

𝑖=𝑘−𝑁+1

𝑒𝑖𝑒T𝑖 𝑘 > 𝑁 (14)

𝐿𝑘 ={{{{{{{{{

𝑘 − 1𝑘 𝐿𝑘−1 +

1𝑘 (𝐻𝑘𝑃𝑘,𝑘−1𝐻𝑘T + 𝑅𝑘) 𝑘 ≤ 𝑁

1𝑁𝑘∑

𝑖=𝑘−𝑁+1

(𝐻𝑖𝑃𝑖,𝑖−1𝐻𝑖T + 𝑅𝑖) 𝑘 > 𝑁 (15)

𝑁 is a sliding window, we can adjust the size of𝑁 to achievedifferent requirements for the sensitivity and accuracy of thefilter. �̂�𝑘 is the actual value of the innovation variance, 𝐿𝑘is the theoretical value of the innovation variance. When�̂�𝑘 deviates from 𝐿𝑘 seriously, there are abnormalities in thesystem.

After the above judgment, when the whole system isabnormal; the next hypothesis test is carried out:

H0: There are anomalies in the measurements.H1: There are anomalies in the system model.

The measurement error equation can be expressed asequation (16).

𝑉𝑘 = 𝐻𝑘�̂�𝑘 − 𝑍𝑘 (16)

𝑉𝑘 can reflect the error of observation. We construct theanomaly factor 𝜂 as equation (17)

𝜂 = tr (𝑉𝑘𝑉T𝑘 )tr (∑𝑉𝑘) (17)

where ∑𝑉𝑘 is the covariance matrix of 𝑉𝑘 [12].When 𝜂 > 𝜆, the measurement of navigation system

is abnormal. 𝜆 is the abnormality detection threshold, andthe range is usually taken as [1.0 3.0]. We build the errorfunction 𝐽(𝛼𝑖) as follows.

𝐽 (𝛼𝑖) = ∑( �̂�𝑘 (𝑖, 𝑖)𝐺𝑘,𝑘−1 (𝑖, 𝑖) + 𝛼𝑖𝑅 (𝑖, 𝑖) − 1)

2

(18)

Among them, 𝐺𝑘,𝑘−1 = 𝐻𝑘𝑃𝑘,𝑘−1𝐻𝑘T. The necessary andsufficient condition for 𝐽(𝛼𝑖) to get the minimum value is𝛼𝑘𝑖 = (�̂�𝑘(𝑖, 𝑖) − 𝐺𝑘,𝑘−1(𝑖, 𝑖))/𝑅𝑘(𝑖, 𝑖). Then we construct theweighted factors as follows:

𝛼𝑘 = diag (𝛼𝑘1 , 𝛼𝑘2 ⋅ ⋅ ⋅ 𝛼𝑘𝑛)𝛽𝑘 = 𝐼𝑚×𝑚

(19)

Among them, 𝑛 is the observation dimension and 𝑚 is thestate variable dimension.

Otherwise, we regard the anomaly of navigation system asmodel anomaly and establish error function 𝐽(𝛽𝑖) as follows.

𝐽 (𝛽𝑖) = ∑( �̂�𝑘 (𝑖, 𝑖)𝛽𝑖𝐺𝑘,𝑘−1 (𝑖, 𝑖) + 𝑅𝑘 (𝑖, 𝑖) − 1)2

(20)

4 Mathematical Problems in Engineering

Table 1: Navigation information error data statistics.

Statistics Filteringmethod 𝑉𝑒/(m/s) 𝑉𝑛/(m/s) 𝑉𝑢/(m/s) L/m 𝜆 /m h /m

Standarddeviation

Standard KF 0.289 0.277 0.329 8.429 6.885 7.842IOAF 0.277 0.287 0.283 7.237 7.177 7.788CWAF 0.166 0.144 0.213 5.319 5.177 7.211

100038

1500

36 109.5

Hei

ght (

m)

109

2000

Simulated flight path

34 108.5

2500

10832 107.510730 106.5latitude (∘) longitude (∘)

Figure 1: Flight path of the aircraft.

The necessary condition for 𝐽(𝛽𝑖) to get theminimumvalue is𝛽𝑘𝑖 = (�̂�𝑘(𝑖, 𝑖)−𝑅𝑘(𝑖, 𝑖))/𝐺𝑘,𝑘−1(𝑖, 𝑖). To ensure the consistencyof the matrix dimensions during the calculation of the filtergain, the weighting factors are constructed as follows:

𝛼𝑘 = 𝐼𝑛×𝑛𝛽𝑘 = diag (𝛽𝑘1 , 𝛽𝑘2 ⋅ ⋅ ⋅ 𝛽𝑘𝑛 , 1 ⋅ ⋅ ⋅ 1)𝑚×𝑚

(21)

4. Simulation Experiment and Analysis

The simulation parameters are set as follows:The initial position of the aircraft is the north latitude

31.9∘, the east longitude 106.5∘ and the height 1000m. Theinitial speed is 100m/s. The yaw angle, pitch angle, and theroll angle are 30∘, 0∘, 0∘ respectively. The error gyro constantdrift is 0.1∘/h(1 𝜎) and the accelerometer bias is 100𝑢𝑔 (1 𝜎).The initial position error is 10m, the speed error is 0.8m/s, andthe output frequency of SINS is 100Hz.The root mean squareof BDS position information error is 20m, the root meansquare of velocity information error is 0.5m/s, the outputperiod is 1s, and the simulation time is 1800s. In order tofully verify the influence of the algorithm on the integratednavigation performance, the flight status of the aircraft shouldinclude climbing, dive, acceleration, cruise and steering. Thesimulation flight trajectory is shown in Figure 1.

In order to verify the performance of the classifiedweighted adaptive filter (CWAF) proposed in this paper, itis assumed that both model and measurement anomaliesexist in INS/BDS integrated navigation system during the

simulation period. In the simulation time of 500s-520s, weset up anomalous measurements. In the time of 500s-510s,the mean square root of velocity measurement errors ineast, north and up directions are 2m/s, -3m/s and 2m/srespectively. In 510s-520s, the velocity errors are 3m/s, -1m/s and-3m/s respectively. We set abnormal model errorsin 1000s-1020s. For convenience of observation, we set theroot mean square of one-step prediction position errorand velocity error caused by abnormal system model as2m, 2m, -2m, 0.2m/s, 0.2m/s and -0.2m/s, respectively. Inthis paper, the standard Kalman filter and the innovationorthogonality adaptive filtering algorithm(IOAF) proposedin [5] are used for simultaneous simulation. The simulationresults are shown in Figures 2–7.

It can be seen from Figures 2–7 that the standard KFhas large fluctuations when the measurement is abnormalduring the 500s-520s period, and the fluctuation amplitudeof the velocity and position errors reaches 2.2m/s and 38m.Compared with the standard KF, the IOAF and the CWAFproposed in this paper have better fault tolerance. Thevelocity and position errors are more stable and keepingwithin 1m/s and 22m respectively. When the model isabnormal during the 1000s-1020s period, the filtering errorsof the standard KF and the IOAF fluctuate greatly. The erroramplitude of standard KF reach 2m/s and 36m.The IOAF haspoor fault tolerance, and the errors reach 2.3m/s and 38m.In contrast, the CWAF can effectively suppress the influenceof model anomalies on the filtering solution. The velocityand position errors are controlled within 0.9m/s and 20m.Navigation information error statistics are shown in Table 1.

Mathematical Problems in Engineering 5

Standard KFIOAFCWAF

200 400 600 800 1000 1200 1400 1600 18000Time (s)

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Eastw

ard

erro

r (m

/s)

Figure 2: Eastward velocity error comparison.

Standard KFIOAFCWAF

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Nor

thw

ard

erro

r (m

/s)

200 400 600 800 1000 1200 1400 1600 18000Time (s)

Figure 3: Northward velocity error comparison.

In Table 1, we compare the standard deviation of velocityand position error obtained by using three filtering methodsin the presence of uncertain anomalies. Through analysis, itcan be seen that the proposed CWAF has better fault-toleranteffect on the whole. And it can obtain higher precisionnavigation information than the standard KF and the IOAFwhen the anomaly type of navigation system is unknown dueto uncertainties. The accuracy of velocity and position are42% and 24% higher than that of standard KF respectively,38% and 22% higher than that of IOAF; the error results aremore stable.

5. Conclusions

Aiming at the problem that the filtering accuracy andstability of SINS/BDS integrated navigation system decreasedue to uncertain model and observation anomalies, a newSINS/BDS integrated navigation method based on classified

Standard KFIOAFCWAF

400 600 800200 1000 1200 1400 1600 18000−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Upw

ard

erro

r (m

/s)

Time (s)

Figure 4: Upward velocity error Comparison.

Standard KFIOAFCWAF

−40

−30

−20

−10

0

10

20

30

40

50La

titud

e err

or (m

)

200 400 600 800 1000 1200 1400 1600 18000Time (s)

Figure 5: Latitude error comparison.

weighted adaptive filtering is proposed. Firstly, the types ofanomalies in navigation system are distinguished by thismethod. Then, based on the analysis of the influence ofmodel anomaly and observation anomaly on navigationfiltering solution, different weighting methods are adoptedfor two types of system anomalies to improve the accuracyand adaptability of integrated navigation system. Finally,the simulation results show that the proposed method caneffectively improve the fault-tolerant performance of inte-grated navigation system in complex environments withunknown types of anomalies. When both model and obser-vation anomalies exist, the velocity and position accuracyof SINS/BDS integrated navigation system are improved by42% and 24% compared with the standard KF, 38% and 22%compared with the innovation orthogonal adaptive filtering.It greatly improves the accuracy of SINS/BDS integrated

6 Mathematical Problems in Engineering

Standard KFIOAFCWAF

200 400 600 800 1000 1200 1400 1600 18000Time (s)

−20

−10

0

10

20

30

40

50

Long

itude

erro

r (m

)

Figure 6: Longitude error comparison.

Standard KFIOAFCWAF

−40

−30

−20

−10

0

10

20

30

40

Hei

ght e

rror

(m)

200 400 600 800 1000 1200 1400 1600 18000Time (s)

Figure 7: Height error comparison.

navigation system and has high fault-tolerance and feasibilityin complex environment with uncertainties.

Data Availability

The data used to support the findings of this study areincluded within the article.

Conflicts of Interest

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

Acknowledgments

This study was co-supported by the National Natural ScienceFoundation of China (Grant no. 61703424).

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