Lever Arm Compensation for Integrated Navigation System of Land Vehicles

Jaewon Seo, Student Member, IEEE, Jang Gyu Lee, Member, IEEE, Chan Gook Park, Member, IEEE

Abstract—For more accurate navigation, the lever arm compensation is considered. The compensation method for GPSand an odometer is introduced and improved compensationmethods are proposed for an odometer to consider the effect ofcoordinate transformation errors and the scale factor error.The methods are applied to an integrated navigation system and experimental result shows the effectiveness.

I. INTRODUCTION

OR intelligent transportation system, the navigation of avehicle is an essential technology. The navigation is

defined as all the related theories and technologies for obtaining position, velocity and attitude of a vehicle. Withthis navigation technology, the transportation vehicle can know its position and can plan trajectory for the destinations. Nowadays, especially, the need for the navigation of landvehicles has increased. In the near future, the ubiquitouspersonal navigation will be spread.

In general, the GPS/INS integrated system is preferred. The Global Positioning System (GPS) is a satellite-basedradio navigation system [1]. It allows a user with a receiver toobtain accurate position information anywhere on the globe. It provides position information disturbed with errors thatwould not increase with respect to time. However, the signalfrom GPS satellite cannot often arrive at a receiver in urban area. In those cases, it cannot give any position information.Inertial Navigation System (INS) is a self-contained,hardly-disturbed, dead-reckoning navigation system [2]-[4].It works according to the inertial principle, and it integrates the inertial measurement from inertial sensors. The accuracy of the INS depends on the ability of the inertial sensors. Thebiases and drifts of the sensors are accumulated in the navigation solution, therefore the navigation error of INS would increase unlimitedly with respect to time. Therefore, in often cases, an odometer is incorporated into INS forreduction of unlimited error. These complementary

properties of GPS and INS motivate the integration system.The GPS/INS integrated system entails the Kalman filter forerror estimation and compensation. Considering the cost and performance, it is most suitable for land vehicles and theresearch for that application is necessary.

Manuscript received February 28, 2005. This work was supported in part by Electronics and Telecommunications Research Institute and Brain Korea 21 program in Korea.

Jaewon Seo is with the School of Electrical Engineering and ComputerScience, Seoul National University, Seoul, Korea (phone: 822-880-7317;fax: 822-878-8198; e-mail: jwseo1@snu.ac.kr).

Jang Gyu Lee is with the School of Electrical Engineering and ComputerScience, Seoul National University, Seoul, Korea (e-mail: jgl@snu.ac.kr).

Chan Gook Park is with the School of Mechanical and AerospaceEngineering, Seoul National University, Seoul, Korea (e-mail:chanpark@snu.ac.kr).

However, in integrating the GPS, INS, and the othersensors, such as an odometer and so on, there are someobstacles. Among them, the lever arm effect is addressed inthis paper. Physically, the sensors of INS, called InertialMeasurement Unit (IMU), the GPS antennas, and theodometers cannot be located at the same position. The lever arm is the difference between the sensing points of thesensors. The position calculated by INS and the position byGPS are different by the length of lever arm. In the samemanner, the velocity of IMU and that of an odometer aredifferent by the lever arm. Therefore, the integration mustconsider the effect of lever arm. In land vehicles, the situationis same. The GPS antenna is usually mounted on the outsideof the vehicles, such as on the roof, while the IMU is locatedinside or at a different spot. The odometer is usually installednear of the wheel of vehicles. Therefore, the information by the GPS, IMU, and odometer must be integrated in systematicapproach. This systematic approach is called lever armcompensation. It makes the navigation solution more accurate. In [3], the lever arm effect and compensation are briefly explained. The lever arm effect of accelerometers, as well asits application to gravimetry is illustrated in [4]. Estimation ofthe measurement error of lever arm length and the misalignment errors of GPS antenna array is treated in [5]. Inthis paper, the lever arm compensation is applied to landvehicle navigation and its experimental results are presented.In addition, an improved one for an odometer consideringcoordinate transformation errors is proposed.

The next section, the general GPS, INS and theirintegration with an odometer is explained. The lever armeffect and compensation method is presented and finally theexperimental results are given.

II. GPS/INS/ODOMETER INTEGRATED SYSTEM

GPS is a satellite-based radio navigation system. Itprovides position of a vehicle. For position calculation, atleast four satellites must be visible generally. However, in theurban area, the number of visible satellites is often less thanfour due to tall buildings and trees. Therefore, with only GPS receiver, the continuous navigation for land vehicles is

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Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

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impossible. On the other hand, INS is operates based on theinertial principle. In the system, the accelerations measuredby accelerometers and the angular velocities by gyroscopes are integrated without external aids. Through the integration,linear velocity, position, and attitude can be obtained.Therefore, it can provide navigation solution continuously.However, the measurement errors in accelerometers andgyroscopes must be integrated together with the true values,and then they accumulate in position solution without limit.GPS/INS/odometer integrated system incorporates thecharacteristics of each navigation system. It integrates the output of IMU at more frequent period and if the externalinformation by GPS or an odometer is available at someinstant, it combines those through some estimation methodsuch as Kalman filtering.

A. Mechanization EquationsThe mechanization equations for inertial navigation are

D

t

E

m

N

Vh

LhRV

l

hRV

L

cos)( (1)

)).((*21

)2(

nen

nie

bn

bib

nen

nie

bnbDEN

Cqq

gVfCVVVV

ThlL is position vector inearth-centered-earth-fixed (ECEF) geodetic frame,

is velocity vector in the local levelnavigation frame, q is the quaternion which is for attitudecalculation, are the meridian and traverse radii of curvature of the earth, is direction cosine matrix from thebody frame to the navigation frame, is accelerationvector measured by IMU in the body frame,

TDEN VVV

tm RR ,nbC

bf

g is the gravityvector, is angular velocity vector measured by IMU inthe body frame, is the earth rate resolved in thenavigation frame and is the transport rate. Therelationships between the quaternion and the direction cosinematrix, and between the Euler angles and the direction cosinematrix are given as

bib

nie

nen

23

22

21

20

1032

2031

10322031

23

22

21

203021

30212

32

22

12

0

)(2)(2

)(2)(2)(2

)(2

qqqqqqqq

qqqq

qqqqqqqqqqqqqqqq

qqqqqqqq

C nb

.coscos

cossinsinsincossinsincossincos

cossinsincoscossinsinsinsincossincoscossinsincoscos

EN

DNDEN

DNDEN

ENE

DNDENDE

DNDENDE

(2)

Detail explanations of (1) and (2) are provided in [2]-[4],[6].

B. Error Model With (1) and the linear perturbation method, following

error model is derived [2], [6].

)(

0000

0000

)(

0000000000

00000)()()()()(

3333

3333

33

33

3333

3333333333

3333333333

3533333231

3324232221

3333331211

twC

CtxFFFF

FFFFFF

twtGtxtFtx

nb

nb

I

III

(3)

zyxzyxa

DENDENf

TafI

x

VVVhlLx

xxtx )((4)

Notation indicates that the variable is an error variableso that, for example, L is the latitude error derived by theperturbation. ’s are accelerometer biases and ’s are gyroscope drifts.

DEN is attitude error represented

by Euler angles. is assumed to be a white Gaussianprocess due to measurement noise of inertial sensors and each elements in the system matrix can be derived by theperturbation [2], [6]. For the GPS position measurement and the velocity measurement of an odometer, the followingmeasurement models are given [2], [6].

)(tw

)()(00)()(0

633333

12333

kodokIodo

kGPSkIGPS

tvtxVIz

tvtxIz (5)

GPSz is position measurement of GPS, is velocityodoz

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measurement by the odometer, and are measurement noises of GPS and the odometer, respectively.They are white Gaussian noises. Subscript k of time

GPSv odov

tindicates the instant of acquisition of the measurement.

C. Kalman Filter With (3)-(5), the Kalman filter can be designed as

followings for navigation error estimation. Kalman filter isoptimal linear filter [7]. After discretization of (3), discreteKalman filter can be constructed. The equations of discreteKalman filter are given;

kkkk

kkkkkk

kTkkk

Tkkk

Tkkk

Tkkkk

kkk

PHKIP

xHzKxx

RHPHHPK

GQGPP

xx

)(

)ˆ(ˆˆ)(

ˆˆ

11

1

(6)

kis state transition matrix of (3), is state covariance

matrix,kP

kG is discretized input matrix, is covariancematrix of , is observation matrix, and is covariance matrix of or . With (1)-(6), the error canbe estimated and GPS/INS/odometer integrated system canbe constructed. In the next section, the lever armcompensation will be described.

kQ

)(tw kH kR

GPSv odov

III. LEVER ARM EFFECT & COMPENSATION

The lever arm effect can be compensated by considering it in the derivation of indirect measurement. The derivationswhich take the effect into account are following.

A. GPS Case The position measurement of GPS can be expressed as

following;

.100

0)cos)/((1000)/(1

LhR

hR

vrCPz

t

m

GPSbnbIMUGPS

(7)

The latitude, longitude, and height are abbreviated with P .The subscript IMU indicates that it is the position of thecenter of IMU and so does GPS. is the offset from IMU to GPS antenna resolved in the body frame. The calculatedposition of the same point will be given as

br

.ˆˆˆ bnbIMUGPS rCPz (8)

The residual for indirect measurement, used to drive the

Kalman filter, is

.00

)(

ˆˆˆ

633333 GPSIbnb

GPSbnbIMU

GPSbnbIMU

GPSbnbb

nbIMU

GPSbnbIMUb

nbIMUGPSGPS

vxrCI

vrCP

vrCP

vrCrCIP

vrCPrCPzz

(9)

P is the position error and .TDEN

B. Odometer Case The velocity measurement of the odometer is

.odonnnbIMUodo vrVz (10)

nnb

is angular rate of the body frame, in which thevelocity is measured, with respect to the navigation frame,expressed in the navigation frame. is the lever armrepresented in the navigation frame. The calculated velocityof the odometer is

nr

.ˆˆˆ nnnbIMUodo rVz (11)

The residual for indirect measurement is

.00

)()(

~ˆ

ˆˆˆ

333333

odo

Ibnbb

bnb

nb

odobnbb

bnb

nbIMU

odobbnb

nb

bbnb

nbIMU

odobbnb

nbb

bnb

nbIMU

odonnnbIMUn

nnbIMUodoodo

v

xrCrCI

vrCrCV

vrC

rCIV

vrCrCV

vrVrVzz

(12)

C. Odometer Case Considering CoordinateTransformation Errors In most cases, the odometer measures the velocity in the

body frame. Therefore, it must be transformed into thenavigation frame for estimation process. However, in theprevious section, the errors in coordinate transformationoperation, which is corresponding to attitude errors, are not considered. In this section, that effect is incorporated into thederivation of indirect measurement. The velocitymeasurement from the body frame can be expressed as,

.ˆodo

bodo

nbodo vVCz (13)

bodoV is the velocity of the odometer in the body frame. The

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calculated velocity of the odometer is the same as (11). The residual is derived as following.

odob

odonb

bnbb

bnb

nbIMU

odob

odonb

bbnb

nbIMUIMU

odob

odonbb

bnb

nbIMU

odob

odonbn

nnbIMUodoodo

vVC

rCrCV

vVCI

rCIVV

vVCrCV

vVCrVzz

)(

)()(

ˆ~ˆˆ

ˆˆˆˆ

odoIbnbIMU

odobnbIMUIMU

odob

odonb

bnbb

bnb

nbIMU

vxrCVI

vrCVV

vVC

rCrCV

333333 00

(14)

D. Odometer Case Considering CoordinateTransformation Errors and Scale Factor Error The odometer produces only pulses which corresponding

to moving distance. The velocity can be calculated with these pulses and a scale factor. A scale factor converts the numberof pulses to velocity which is moving distance per time of unit.In this case, the error of the scale factor must exist. Therefore, some sophisticated navigation systems employ the estimationmethod for the scale factor error. In that system, the errormodel becomes different slightly. (15) and (16) are the errormodel including the scale factor error. is the scale factor error of the odometer.

xok

)(

0000

0000

)(

00000000000000000000

)()()()()(

3434

3333

33

33

3333

143434343434

133333333333

133533333231

133324232221

133333331211

twC

C

txFFFF

FFFFFF

twtGtxtFtx

nb

nb

I

III

(15)

zyxzyxa

DENDENf

TafI

x

VVVhlLx

xxtx )(

(16)

The scale factor error is assumed to be random constant

bias in the above system. If the error is considered in thenavigation system, the lever arm compensation methodbecomes different, too. The velocity measurementconsidering the error is given as,

.)1(ˆodo

bodoxo

nbodo vVkCz (17)

The calculated velocity of the odometer is the same as (11). The residual is derived as following.

odoInnnbIMUb

nb

IMU

odoxonnnbIMU

bnbIMUIMU

odoxonnnbIMU

bodo

nb

bnbb

bnb

nbIMU

odonnnbIMUxo

bodo

nb

bnbb

bnb

nbIMU

odob

odoxonb

bbnb

nbIMUIMU

odob

odoxonbb

bnb

nbIMU

odob

odoxonbn

nnbIMUodoodo

vxrVrC

VI

vkrV

rCVV

vkrVVC

rCrCV

vrVkVC

rCrCV

vVkCI

rCIVV

vVkCrCV

vVkCrVzz

)(

00)(

)(

)(

)1()(

)()(

)1(ˆ~ˆˆ)1(ˆˆˆˆ

333333

(18)

All above cases made the linear relationships between theerror states and indirect measurements and they can be easilyapplied to Kalman filter structure.

IV. EXPERIMENTAL RESULT

The derived lever arm compensation methods have been tested on the land vehicle. It is a van equipped with IMU,GPS antenna, and an odometer. The IMU is LN-200produced by Northrop Grumman Corporation. It uses fiberoptic gyros (FOGs) and silicon accelerometers (SiAc's) for measurement of vehicle angular rate and linear acceleration. The detail characteristics of LN-200 are presented in [8]. It isinstalled at the center of the roof of the van. The GPS antennais also mounted on the roof, but is located the left side of theLN-200. The odometer is installed at the left rear wheel. SeeFig. 1. The lever arm vectors from the origin of the bodyframe, which is the center of IMU, were measured and they are in table 1.

The trajectory is illustrated in Fig. 2. The van started in S,went to east direction, turned left, went to north direction,turned left once more, and then finally went to west directionstraightly. The star marks indicate the position measurementof GPS. In Fig. 3 (a), (b), and (c), the lever arm compensation

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effects are illustrated. As explained above, the star marks are the measurement of GPS and the dotted line is the navigationresult produced without any lever arm compensation. Thedashed line is the result with the lever arm compensationmethod for the case D in the previous section. The solid line istrue trajectory of the center of IMU. Since the antenna is mounted at the left side of IMU, the measurement of GPS isbiased to the negative direction of Y axis in the body framewith respect to IMU. Compared to the result without themethod, the navigation result with the lever armcompensation is shifted to the Y axis direction in the bodyframe and it makes sense physically.

V. CONCLUSION

The lever arm compensation for land vehicles is addressed, and the compensation method for the odometer consideringcoordinate transformation errors is proposed. It is applied toGPS/INS/odometer integrated system and shows improvedresult.

The compensation method is simple, easy to apply, and useful for accurate navigation of land vehicles. It is not only for land vehicles, but also for any other transportation system,such as airplanes, ships if they are equipped with any positionand velocity sensor as well as GPS and an odometer.

ACKNOWLEDGMENT

The authors appreciate the help of Electronics andTelecommunications Research Institute for experiment.

TABLE ITHE LEVER ARM VECTORS IN THE BODY FRAME

Sensor X Y Z

GPS Antenna -5cm -69cm -11cm

odometer -48cm -71cm 173cm

Fig. 1. The land vehicle for test.

4060 4080 4100 4120 4140 4160 4180 4200 4220 4240 42601000

1050

1100

1150

1200

1250

1300

1350

1400

East(m)

Nor

th(m

)

S

F

Fig. 2. The trajectory for test.

4190 4195 4200 4205 4210 4215 4220 4225 42301040

1045

1050

1055

East(m)

Nor

th(m

)

S

Fig. 3. (a) The improved navigation result of the lever arm compensation.

4240 4245 4250 42551100

1120

1140

1160

1180

1200

1220

1240

1260

1280

1300

East(m)

Nor

th(m

)

Fig. 3. (b) The improved navigation result of the lever arm compensation.

IMU (LN-200)

Odometer

GPS antenna

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4080 4090 4100 4110 4120 4130 4140 4150 4160 4170 41801370

1375

1380

1385

East(m)

Nor

th(m

)

Fig. 3. (c) The improved navigation result of the lever arm compensation.

REFERENCES

[1] E. D. Kaplan, Understanding GPS principles and applications. Artech House, 1996.

[2] D. H. Titterton and J. L. Weston, Strapdown inertial navigation technology. Peter Peregrinus Ltd., 1997.

[3] J. A. Farrell and M. Barth, The global positioning system & inertial navigation. McGraw-Hill, 1999.

[4] C. Jekeli, Inertial navigation systems with geodetic applications.Walter de Gruyter GmbH & Co., 2000

[5] S. Hong, M. H. Lee, S. H. Kwon, and H. H. Chun, “A car test for the estimation of GPS/INS alignment errors,” IEEE Trans. Intell.Transport. Syst., vol. 5, pp. 208–218, Sep. 2004.

[6] J. G. Lee, et al., “Development of PIG navigation system,” Tech. Rep., Automation and Systems Research Institute, Seoul National University, 2002.

[7] R. G. Brown and P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering. John Wiley & Sons, 1997.

[8] LN-200 Description, Northrop Grumman Corporation,http://www.nsd.es.northropgrumman.com/Html/LN-200/

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