improvement of accuracy in aircraft navigation by data fusion technique

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –

6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 8, August (2014), pp. 35-46 © IAEME

35

IMPROVEMENT OF ACCURACY IN AIRCRAFT NAVIGATION BY DATA

FUSION TECHNIQUE

Mahasweta Bhattacharya1

1(Electronics and Communication Engineering, West Bengal University of Technology,

West Bengal, India)

ABSTRACT

Navigation of a aircraft is to reach a destination accurately and with minimum diversions.

This needs to guide the aircraft on the desired track. For a fighter aircraft which needs fast

manoeuvring to achieve the mission objectives need continuous and accurate position, attitude and

velocity updates.

The sensors used in an aircraft are GPS and Inertial Navigation System. The Inertial

Navigation System used in aircraft navigation provides parametric information in terms of

position(latitude, longitude and altitude), velocity(Velocity along north, east and vertical direction)

and attitudes(Pitch, roll, yaw). INS is working based on three gyroscopes and three accelerometers.

GPS works based on the satellite information of the aircraft position.

The update rate of INS solutions is high but in course of time, the accuracy of the

navigational solution degrades. GPS provides highly accurate position and velocity solutions but at a

slow update rate. Hence standalone use of either one for obtaining navigational solution is not

employed. This is a major problem in military navigation where highly accurate data with fast update

rate is necessary.

The solution to this problem can be obtained by designing a navigational algorithm which

performs sensor data fusion using the data from INS and GPS of an aircraft by means of a Kalman

Filter. In this project, a sensor data fusion algorithm for performing sensor fusion of INS and GPS

using 9th order Kalman filter has been developed. An initial estimation of the navigational

parameters that were needed to be corrected was taken. A process model using INS data and

measurement model using GPS data is formed. Using the process model, measurement model and

the Kalman filter, the errors in the INS data were corrected. The corrected navigational data thus

obtained is then compared with a standard system hybrid navigational data.

Keywords: INS, GPS, Kalman Filter, Estimation, Accuracy.

INTERNATIONAL JOURNAL OF ADVANCED RESEARCH

IN ENGINEERING AND TECHNOLOGY (IJARET)

ISSN 0976 - 6480 (Print)

ISSN 0976 - 6499 (Online)

Volume 5, Issue 8, August (2014), pp. 35-46

© IAEME: http://www.iaeme.com/IJARET.asp

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36

I. INTRODUCTION

Integration of INS and GPS has been indispensable because of the inherent drawbacks of the

individual system. INS has its inherent errors that gradually accumulate to finally give highly

erroneous data. GPS updates at a very slower rate of about 1sec. Both the drawbacks are

disadvantageous for successful military navigation. In general, an INS has a high update rate of

about 20ms and GPS has high accuracy in measurement. Thus using any one of them as a standalone

system hinders accurate and faster computation. Thus integration is needed.

If the errors in INS are not corrected then the error will go on accumulating thereby giving a

final data which will be highly erroneous and faulty. Thus the INS position and velocity data must be

corrected.

The main objective of the project was to improve the accuracy in measurement of a

navigational system by fusing the data coming from both INS and GPS so that an integrated system

through the fused data can be used in aircrafts where precise and fast computations are needed. To

accomplish this objective the following steps are involved:

• Design and develop the process model based on INS data

• Designing the measurement model based on GPS data

• Making an initial estimation based on the process model

• Feeding the estimation and measured data to Kalman Filter which minimizes the error between

the corrected and the actual data and provides a corrected estimate for the next state.

• The processes are continued in recursion for the whole set of data to arrive at a final corrected

set of data.

For testing the developed sensor data fusion algorithm, it has been coded in MATLAB. A set

of data relating to INS and GPS of a particular flight sortie of aircraft has been taken. The INS data

and the GPS data has been fused using the algorithm to give a corrected hybrid output. The algorithm

has been tested by comparing the computed hybrid data with the system hybrid data obtained from

INGPS system installed on the aircraft. It has been found that the error between computed hybrid

data and system hybrid data was very minimal and thereby establishing the success of the algorithm.

II. ERRORS OBSERVED IN SAMPLE INS DATA

As mentioned earlier, navigational data of a specific aircraft system has been taken as sample

in our project. The errors in the acceleration of the INS data were observed as such. The three plots

in the figure below show the INS acceleration measured by accelerometers. The plots show how

much the reading of the INS is full of errors. The above plots also are erroneous.

Errors in the accelerations and angular rates occur due to manufacturing error, biases etc

which lead to steadily growing errors in position and velocity components of the aircraft, due to

integration. These are called navigation errors and there are nine of them - three position errors, three

velocity errors, two attitude errors and one heading error. If an unaided INS is used, these errors

grow with time. It is for this reason that the INS is usually aided with GPS for accurate navigation

solutions.



37

Fig.1: The Accelerometer Reading along x, y and z axis respectively with respect to time

Fig. 2: The three plots above shows the turn angle rates about the three axes

III. ERRORS OBSERVED IN GPS DATA

Although GPS provides accurate navigational data free of errors but the GPS faces several

drawbacks. Firstly, the system consists of a constellation of fixed satellites which sends signals to

determine position. This takes higher computation time and thus the system has higher update rate.

The data rate is about 1sec. But even faster computation is necessary for military navigation. Thus

standalone GPS is not advantageous. In the system the GPS readings have been taken at t=20ms

which is the sample time for the INS system. We observe staircase type graph for the parameters.

This is because the update rate of GPS is much lower compared to the INS. While INS is updating at

20ms GPS is updating at 1s. Therefore, for 1s GPS keeps at data constant and data is updated after

that. This is the cause of the stairs. The values are updated after 50 such samples. So for 50 updated

values of INS, GPS data remains constant. At such a lower update rate we cannot get the proper

navigational details. Thus GPS also cannot be efficiently used as a standalone navigation system.

The figure below shows the nature of the data obtained from GPS.

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976

6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 8, August

Fig.3

IV. WORKING PRINCIPLE OF KALMAN FILTER

Kalman filter [3],[4],[6] estimates the state vector based on the knowledge of the

measurement model as well as on the knowledge of the process model. The process noise and

measurement noise are assumed to be independent of each other. The Kalman filter is a recursive

algorithm which makes predictions from previous state and then updates the prediction using

measurement to obtain an optimal state vector. The main objective of the Kalman Filter is

estimate of the n-state vector xk denoted by

minimizing the mean squared error between the estimated and the real value. in any system, the next

state is based on the previous state and is related to it through the state transition matrix. Presence

any input noise will affect the output state as well. Thus the process of the system can be modelled

as:

The noise is uncorrelated with respect to time and thus the covariance matr

Therefore,

Again, through the measurement model we can estimate a particular state at that very instant.

Noise in the model will affect the measurement and thus is

model thus can be formulated as:

Similarly, the covariance matrix associated with

objective is to make estimation such that the estimated value should be as close as possible to the

actual value of the erroneous data. Practically, it is difficult to make such close estimation and thus

there may be present some errors in our estimated value and th

so as to bring down that error to a minimum. We denote the error covariance matrix as

. The target is to minimize this error covariance matrix so that our estimation tallies

with the actual data.

Now, the function of a Kalman Filter is divided in two stages. The stages are: a). Updating

and b). Predicting. This stages occur in recursion and in every stage the Kalman Gain


6499(Online) Volume 5, Issue 8, August (2014), pp. 35-46 © IAEME

38

Fig.3: Altitude obtained from GPS

WORKING PRINCIPLE OF KALMAN FILTER



assumed to be independent of each other. The Kalman filter is a recursive


measurement to obtain an optimal state vector. The main objective of the Kalman Filter is

denoted by which is a function of all the previous states and


state is based on the previous state and is related to it through the state transition matrix. Presence


(equation 1)

The noise is uncorrelated with respect to time and thus the covariance matrix associated with

(equation 2)


Noise in the model will affect the measurement and thus is taken into account. The measurement

(equation 3)

Similarly, the covariance matrix associated with is . Now the main

is to make estimation such that the estimated value should be as close as possible to the


there may be present some errors in our estimated value and the actual value. So we use a technique



function of a Kalman Filter is divided in two stages. The stages are: a). Updating

stages occur in recursion and in every stage the Kalman Gain


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assumed to be independent of each other. The Kalman filter is a recursive


measurement to obtain an optimal state vector. The main objective of the Kalman Filter is to find an

which is a function of all the previous states and


state is based on the previous state and is related to it through the state transition matrix. Presence of


(equation 1)

ix associated with is

(equation 2)


taken into account. The measurement

(equation 3)

. Now the main

is to make estimation such that the estimated value should be as close as possible to the


e actual value. So we use a technique



function of a Kalman Filter is divided in two stages. The stages are: a). Updating

stages occur in recursion and in every stage the Kalman Gain is



computed and the state of the system as well as the associated Error C

for every state in the recursive loop. The equations associating with this updating stage are as

follows. (The superscript“-”denotes predicted state (priori state).

The equations that govern the stage of predicting are:

Equation shows that if is large, K

be less weighted. The estimated state will be nearer to predicted state. When K

term will have more weightage so the estimated state will be predicted state + weighted residual.

When Rk approaches zero then measurement more trus

zero then prediction is trusted more.

So in every stage the Kalman Filter narrows down the error between estimated and actual

data, updates the state and then predicts for the next state. This occurs in a

stages available finally giving a data which will match the actual data thereby filtering out the actual

data from the erroneous one.

In a system, an estimate about the next state can be determined using the process model. But

due to inherent error in the process model the estimated value at a particular instant will not be close

to the true value at that instant.

So we take another model known as the measurement model which gives measurement of

some of the parameters at that particu

models and corrects the estimated data.

In our project, the INS serves as the process model and the GPS serves as the measurement model.

V. PROCESS MODEL OBTAINED FROM INS SYSTEM

• Reference Ellipsoidal Model

Fig 3



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computed and the state of the system as well as the associated Error Covariance Matrix is updated


”denotes predicted state (priori state).

(equation 4)

(equation 5)

(equation 6)

The equations that govern the stage of predicting are:

(equation 7)

(equation 8)

is large, Kk will be small. Therefore, the term

be less weighted. The estimated state will be nearer to predicted state. When Kk is large then second


approaches zero then measurement more trusted than prediction. When


data, updates the state and then predicts for the next state. This occurs in a loop for the number of



inherent error in the process model the estimated value at a particular instant will not be close


some of the parameters at that particular instant. The Kalman Filter optimally combines these two

models and corrects the estimated data.


PROCESS MODEL OBTAINED FROM INS SYSTEM

Fig 3: Reference earth model


© IAEME

ovariance Matrix is updated


(equation 4)

(equation 5)

(equation 6)

(equation 7)

(equation 8)

will

is large then second


ted than prediction. When approaches


loop for the number of



inherent error in the process model the estimated value at a particular instant will not be close


lar instant. The Kalman Filter optimally combines these two




Length of the semi-major axis, R=

Length of the semi-minor axis, r=6356752.3142 meters

Eccentricity of the Ellipsoidal Model, e=0.0818191908426

Meridian Radius of Curvature,

Transverse Radius of Curvature,

• Spherical Earth Model

Where normal gravity at height h=0 i.e 9.81m/s

The design of the process model has been explained below.

INS Position error dynamics with respect to position, velocity and attitude are :

INS velocity error dynamics with respect to position, velocity and attitude are:



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major axis, R=6378137.0 meters

minor axis, r=6356752.3142 meters

Eccentricity of the Ellipsoidal Model, e=0.0818191908426

Meridian Radius of Curvature,

Transverse Radius of Curvature,

(equation 9)

normal gravity at height h=0 i.e 9.81m/s2.

The design of the process model has been explained below.

INS Position error dynamics with respect to position, velocity and attitude are :

(equation 10)

(equation 11)

respect to position, velocity and attitude are:

(equation 12)

(equation 13)


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(equation 9)

(equation 10)

(equation 11)

(equation 12)

(equation 13)



=

INS attitude error dynamics with respect

Thus, converting to skew symmetric form we get,

Therefore, the final process model,

Where state vector,

VI. MEASUREMENT MODEL

Measurement model is represented as

Where represent vector measurement of state xk at time tk. The measurement matrix is

represented as:



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= (equation 14)

INS attitude error dynamics with respect to position, velocity and attitude are:

(equation 15)

(equation 16)

(equation 17)

to skew symmetric form we get,

(equation 18)

(equation 12)

(equation 13)

(equation 14)

MEASUREMENT MODEL - GPS ERROR MODELLING

Measurement model is represented as

(equation 15)

represent vector measurement of state xk at time tk. The measurement matrix is


© IAEME

(equation 14)

(equation 15)

(equation 16)

(equation 17)

(equation 18)

(equation 12)

(equation 13)

(equation 14)

(equation 15)

represent vector measurement of state xk at time tk. The measurement matrix is



If INS and GPS are same with the difference in measurement leading to zero then that may

lead to instability of the system. To avoid the instability, first row of equation (6.15) and (6.16) is

multiplied with and second row with

And

VII. DESIGN OF THE KALMAN FILTER AND INS, GPS INTEGRATION

The F-matrix is in continuous time which is discretized as,

Kalman Filter estimates next state from previous state through the Kalman

equation (6.5),

The (k+1)th

state can be estimated according to equation (6.7) as:

The corrected error covariance matrix is given as,

The Kalman Gain is calculated according to equation (6.4) as:

The updated error covariance matrix is given as:



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(equation 16)

(equation 17)



and second row with . Therefore, the equation becomes

(equation 18)

(equation 19)

DESIGN OF THE KALMAN FILTER AND INS, GPS INTEGRATION

matrix is in continuous time which is discretized as,

(equation 20)

Kalman Filter estimates next state from previous state through the Kalman Gain according to

(equation 21)

state can be estimated according to equation (6.7) as:

(equation 22)

The corrected error covariance matrix is given as,

(equation 23)

The Kalman Gain is calculated according to equation (6.4) as:

(equation 24)

The updated error covariance matrix is given as:


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(equation 16)

(equation 17)



. Therefore, the equation becomes

(equation 18)

(equation 19)

DESIGN OF THE KALMAN FILTER AND INS, GPS INTEGRATION

(equation 20)

Gain according to

(equation 21)

(equation 22)

(equation 23)

(equation 24)

(24)



VIII. BLOCK DIAGRAM

Fig. 4: Block Diagram of INGPS integrated circuit using Kalman Filter

The above figure shows the block diagram of the developed navigational algorithm for fusing

the INS and GPS data using Kalman Filter to obtain accurate navigational solution. The INS data in

terms of latitude, longitude, altitude, Vn, Ve and Vd. and the GPS data in terms of latitude, longitude,

altitude, Vn, Ve and Vd are used with 9

states i.e. .latitude, longitude, altitude, Vn, Ve, Vd, roll, pitch and yaw.

This is done by forming the process model using the INS data and the measurement model

using the GPS data. Further Kalman Filter is used for computing the erro

Kalman Gain. Using the process model, measurement model and the Kalman Gain the corrected

error is computed. This computed error is then added to the INS data to obtain the computed hybrid

data. To analyze the performance of the dev

is then compared with the system hybrid data.

IX. RECURSIVE FUNCTION OF THE KALMAN FILTER

Fig. 5: Diagram of the Kalman Filter Recursive Cycle



43

Block Diagram of INGPS integrated circuit using Kalman Filter


Kalman Filter to obtain accurate navigational solution. The INS data in


altitude, Vn, Ve and Vd are used with 9th

order Kalman filter to compute the error in each of the

states i.e. .latitude, longitude, altitude, Vn, Ve, Vd, roll, pitch and yaw.


using the GPS data. Further Kalman Filter is used for computing the error covariance matrix and



data. To analyze the performance of the developed navigational algorithm, the computed hybrid data

is then compared with the system hybrid data.

RECURSIVE FUNCTION OF THE KALMAN FILTER

Diagram of the Kalman Filter Recursive Cycle


© IAEME

Block Diagram of INGPS integrated circuit using Kalman Filter


Kalman Filter to obtain accurate navigational solution. The INS data in


rror in each of the


r covariance matrix and



eloped navigational algorithm, the computed hybrid data



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X. RESULT OBTAINED FROM SAMPLE

Fig. 6: Latitude and Longitude obtained from system and computation of algorithm

Fig. 7: Altitude and Velocity along north axis(Vn) obtained from system and computation of

algorithm

Fig. 7: Velocity along lateral and vertical axis(Ve and Vd) obtained from system and computation of

algorithm



45

Fig. 8: Roll and Pitch obtained from system and computation of algorithm

Fig. 8: Yaw obtained from system and computation of algorithm

XI. ANALYSIS

Table 1: Tabular comparison of error deviation for the 3 samples of the computed data and the

hybrid data

PARAMETERS ERROR STANDARD DEVIATION IN

SAMPLE 1

Latitude, λ (degree) 6.287×10-5

Longitude, µ(degree) 4.378×10-5

Altitude, h(feet) 55.3443

Velocity towards north, Vn (m/s) 3.7069

Velocity towards east, Ve (m/s) 2.6915

Velocity towards the vertical, Vd (m/s) 1.9197

Roll, Φ (degree) 0.2849

Pitch, θ (degree) 0.3369

Yaw, Ψ (degree) 0.1359

Standard deviation of errors for all these parameters were in acceptable range except the

altitude. The above table shows the mean error standard deviation computed using the three sample

data. In altitude a constant offset was observed in the standard error deviation because of the vertical

channel problem. The altitude result sows that the computed data deviates gradually from the system

hybrid data. This is because the system data is making its own correction based on the correction

algorithm installed where all the necessary factors have been considered, whereas, our algorithm has



46

considered only 9 parameters. Hence although in the beginning the two data are quite similar but

gradually deviation increases. Altitude errors can be corrected by introducing the vertical channel

modelling where the gravitational acceleration variation with respect to altitude is corrected by

introducing another observer. In some of the sample solutions noises were observed which led to the

high maximum value of error. This can be further removed by introducing digital smoothers as a next

module after the Kalman Filter module.

XII. CONCLUSION

The main aim of the project was to achieve successful data fusion for improvement of INS

and GPS systems. INS and GPS show their inherent disadvantages. While INS system leads to error

build up at the same time GPS despite its high accuracy, update rate is low. Precision and faster

computation are of utmost importance in military navigation which led to the need to fuse the INS

and the GPS for obtaining accurate navigational solution using INS and GPS. The fusion has been

achieved through Kalman Filter which works on the principle of estimation and update. The 9th

order

Kalman Filter has been used to correct the 9 navigational parameters i.e. latitude, longitude, altitude,

velocities in three axes, roll, pitch and yaw subsequently by forming the process and the

measurement model. Observed algorithm was very good in latitude, longitude, altitude, vn, ve, vd,

roll, pitch and yaw. Further improvement can be made by introducing the accelerometer bias and

gyroscope sensitivity models. Observed errors in altitude and vd can be further minimized by

modelling the vertical channel. By fusion of INS and GPS thus we can aim to achieve accurate

navigation which can be employed in military. Data fusion can also be used in fields of medicine,

space research, economics etc where high precision data is important.

XIII. ACKNOWLEDGEMENT

I would like to offer my heartfelt gratitude to the Mission Combat and System Research and

Design Centre Department, Hindustan Aeronautics Limited, Bangalore for their guidance and

constant support for the execution of the project.

XIV. REFERENCES

1. David H. Titterton and John L. Weston, Strapdown Inertial Navigation Technology - 2

nd

Edition(2004) The Institution of Electrical Engineers.

2. Elliott D. Kaplan and Christopher J. Hegarty, Understanding GPS Principles and

Applications- 2nd

Edition(2006) Artech House.

3. Mohinder S. Grewal and Angus P. Andrews, Kalman Filter Theory and Practice using

MATLAB-3rd

Edition(2008) John Wiley & Sons, Inc.

4. Paul Zarchan Fundamentals of Kalman Filtering—A Practical Approach, 2nd Edition(2005)

American Institute of Aeronautics and Astronautics, Inc.

5. Robert M. Rogers, Applied Mathematics in Integration Navigation System- 2nd

Edition(2003)

AIAA Education Series.

6. Eun-Hwan Shin, Accuracy Improvement of Low Cost INS/GPS for Land Applications(2001)

UCGE Reports.

7. Deepeshnamdev, Monika Mehra, Prerna Sahariya, Rajeshwaree Parashar and Shikha Singhal,

“Navigation System by using GIS and GPS”, International Journal of Electronics and

Communication Engineering & Technology (IJECET), Volume 4, Issue 3, 2013,

pp. 232 - 243, ISSN Print: 0976- 6464, ISSN Online: 0976 –6472.

improvement of accuracy in aircraft navigation by data fusion technique

Technology

gps data

accurate data

velocity data

fused data

measured data

actual data

afinal data

setof data