GPR ANTENNA POSITION AND ORIENTATION ESTIMATION USINGSTRAPDOWN INERTIAL NAVIGATION

Friedrich Roth, Gary R. OlhoeftDepartment of Geophysics, Colorado School of Mines, Golden, CO 80401, USA

friedrich_roth@yahoo.com golhoeft@mines.edu

ABSTRACT

This paper discusses the possible use of strapdown inertialnavigation for real-time ground penetrating radar (GPR)antenna position and orientation estimation along arbitrarythree-dimensional acquisition lines. Strapdown inertialnavigation theory has been studied extensively in theliterature for aircraft, missile and space navigation. Here, wegive an overview of the theory as it applies to the antennaposition and orientation problem. This includes thedefinition of the relevant coordinate frames and attitudeparameters, a discussion of the measured acceleration andangular velocity, and a description of the four primarycomputational tasks pertinent to strapdown inertialnavigation. These are the initial alignment of the system, theintegration of angular velocity into attitude (attitudeupdating), the acceleration transformation and integrationinto velocity (velocity updating) and the integration ofvelocity into position (position updating). The key elementsof using a low-grade versus a high-grade inertialmeasurement unit (IMU) are pointed out. The actualperformance of a commercially available low-grade IMU isevaluated based on a series of navigation experiments. Theexperimental results show that the tested IMU is far frombeing accurate enough for completely self-containedantenna positioning and that the precise calibration for scalefactors, biases and axis misalignments is vital. The observedorientation accuracy (error of less than 1 degree after 60seconds) suggests the integration of the tested IMU withodometry, extending the applicabili ty of the latter toenvironments with topography or where changing of theprofile direction due to obstacles is necessary. Anotherpossible use of low-grade IMUs might be for moresophisticated “rubber sheeting” techniques.

Key words: strapdown inertial navigation, position,orientation, inertial measurement unit (IMU)

INTRODUCTION

Ground Penetrating Radar is used for the investigation ofthe subsurface or man-made structures in a variety ofenvironments and settings. There is need for flexible and

accurate antenna positioning and, to use polarizationinformation, antenna orientation sensing. Conventionalantenna positioning techniques are generally limited in theirapplicabili ty. For example, wheel and string odometers donot allow for topography, “ rubber sheeting” (Bochicchio,1988) requires a smoothly varying topography and a nearlyconstant antenna speed for accurate positioning, lasertheodolite surveying can only be used in areas with noobstacles and is slow unless a self-tracking theodolite isemployed, and real-time kinematic GPS can only be used atlarge open outdoor sites. In addition, none of thesetechniques provide antenna orientation information.

Strapdown inertial navigation is potentiall y able to provideantenna position as well as orientation in real-time alongarbitrary three-dimensional acquisition lines and allows fastdata acquisition. Since strapdown inertial navigation iscompletely self-contained, i.e. it does not rely on otherdevices such as beacons or satellit es, it can be used invirtually any environment, including indoors.

THEORY

The idea on which inertial navigation is based is essentiallythat of dead reckoning, where continuous information ofspeed and direction of motion of an object is used to updateits position. To keep track of the object’s motion, strapdowninertial navigation systems use a cluster of threeaccelerometers and three gyroscopes attached rigidly to theobject. The accelerometer-gyro cluster is commonlyreferred to as inertial measurement unit (IMU). Theaccelerometers sense the object’s acceleration which whenintegrated once yields velocity, and integrating velocityyields distance traveled. The gyroscopes measure angularvelocity thereby providing the necessary information forknowing where the accelerations are directed.

Notation

This section gives a brief overview of the notationalfeatures specific to the discussion of the theory. In the

following, F, 1F and 2F are arbitrary coordinate frames, v isan arbitrary vector and ( ) is an arbitrary quantity.

=coordinate frame transformation matrix relatingframe 2F to frame 1F

=attitude quaternion describing the orientation offrame 2F relative to frame 1F

Fv =column matrix with elements equal to theprojection of v on the axes of frame F

F~v =skew-symmetric form of Fv , defined as the matrix

���

��

�����

� ��0vv

v0v

vv0

FF

FF

FF

xy

xz

yz

where FFF zyx v,v,v are the elements of Fv

=equivalent vector quaternion of Fv , having avector part equal to Fv and a zero scalar part

21FF� =angular velocity of frame 2F relative to frame 1F

=first time derivative of quantity= second time derivative of quantity

Coordinate Frames

From a theoretical point of view, inertial space serves as thereference for the measurements of accelerometers andgyroscopes. For practical purposes, inertial space may bedefined through a coordinate frame having its origin at theearth’s center of mass and that is non-rotating relative to thefixed stars (Britting, 1971). The orientation of its coordinateaxes is arbitrary. Particular orientations are discussed inChatfield (1997) and Roth (1999). In equations, the earth-centered inertial frame is designated by the letter i.

The earth frame, designated by the letter e, is a coordinateframe whose origin is at the earth’s center of mass and hasaxes fixed to the earth. It is typically defined with one axisparallel to the earth’s rotational axis and with the other axeslying in the equatorial plane (Savage, 1998). The earthframe rotates relative to the earth-centered inertial frame atthe earth’s rotation rate:

s/rad10292115.7 5�ie

� (Chatfield, 1997). (1)

A local navigation frame needs to be defined which is usedas a reference for the position and orientation of the antennaand describes the space associated with the GPR surveyarea in a way that is suitable for the interpretation of radardata. These objectives are met by the local level frame(Roth, 1999) designated by the letter l (fig. 1). Its origin islocated at some suitable reference point, e.g. the startingpoint of an acquisition line. Its z axis points in the directionof the local gravity vector g, its x axis runs parallel to the

projection of a user chosen reference line onto thehorizontal plane, and the y axis completes the set. Thereference line can be any line connecting two points in thesurvey area.

The ideal IMU would have accelerometer and gyroscopeinput axes that constitute a perfectly orthogonal coordinateframe and accelerometers that all measure at the same point.Of course during manufacture perfect physical alignmentcan never be achieved and accelerometers have finitedimensions. The ideal IMU is reflected in the cube frame(Roth, 1999) designated by the letter c (fig. 1). Its origin liesat the center of the accelerometer cluster. Its axes can bedefined to be the orthogonalized accelerometer input axes.Preferably, if the IMU was calibrated, the cube frame axesare specified by the axes of the calibration table, which isgenerally a two-axis rotation table (Stave, 1996).

The non-orthogonality (axis misalignment) of the inertialsensors (accelerometers and gyroscopes) is accounted for bythe definition of the accelerometer and gyroscope frames(Chatfield, 1997; Roth 1999). Their origins coincide withthat of the cube frame and their axes are specified by theinput axes of the corresponding inertial sensors.

Attitude Parameters

The orientation of the antenna is identified with that of thecube frame relative to the local level frame. This orientationmay be described by the yaw, pitch and roll angles, theattitude quaternion lcQ , or directly by the direction cosinematrix , which transforms vectors from their cube framerepresentation to their local level frame representation.

The yaw ( � ), pitch ( ) and roll ( � ) angles describe threeconsecutive rotations, whose order is important and thatwould rotate the local level frame into the cube frame(Etkin, 1959) as depicted in figure 1. The yaw angledescribes the azimuthal orientation of the antenna and thepitch and roll angles describe its tilt.

The attitude quaternion essentiall y describes a singlerotation that would rotate the local level frame into the cubeframe. Given that the latter is rotated with respect to the firstthrough angle � about the unit vector u (fig.1), the attitudequaternion is of the form

� �� � ������ � ��u

Q2/sin

2/coslc (Roth, 1999). (2)

The relationships between the yaw, pitch and roll angles,the attitude quaternion and the direction cosine matrix aredetailed in Kuipers (1999).

1

2

FFC

21FFQ

FQv

.

)(..

)(

lcC

Even though the yaw, pitch and roll angles are moreintuitive than either the attitude quaternion or the directioncosine matrix, it is not advisable to use them as the primaryparameters for the attitude updating (Roth, 1999). They arenot defined for vertical antenna orientations ( deg90

���).

In the differential equations describing their rates of change,vertical orientations are associated with singularities, whichintroduce instabili ty in the updating when approaching thevertical state. The attitude quaternion and the directioncosine matrix behave well for arbitrary orientations.Consequently attitude quaternion and direction cosinematrix updating would also work when inspecting sewerwalls, dams, etc. The attitude quaternion and the directioncosine matrix change according to the following differentialequations (Roth, 1999; Savage 1998):

lc

lie

cic

lc

clc

lc

lc CCCC ��� ~~~ � !

(3)

and

" #lclie

ciclc

clclcc QQQQ $%$&$&

QQQ5.05.0l '''(

(4)

where ) denotes quaternion multiplication. Both directioncosine matrix and attitude quaternion updating are widelyused in practice with virtually identical results (Savage,1998).

Despite the shortcomings of the yaw, pitch and roll angles,they are the variables of choice for the initial alignment ofthe navigation system (discussed below), since they allowthe azimuthal orientation of the antenna and its tilt to beconsidered separately.

Measured Quantities

When corrected for axis misalignments and biases, the IMUaccelerometer output is a measure of the specific forceacting along the cube frame axes, denoted by cf . Thespecific force is defined as the difference between theinertial and gravitational acceleration (Britting, 1971). Roth(1999) showed that the specific force acting along the cubeframe axes can be written as

Figure 1: Antenna system orientation and position (symbols are explained in the text)

Using the yaw ( * + , pitch ( , ) and roll (- ) angles to describe theantenna orientation:

Using the attitude quaternion lcQ to describe the antenna

orientation:

u

.

..cy

ly

lz cz

cx

lx

g

ly

lz

lx

0a

initial antenna position

trajectory

antennaposition

antennaIMU

d

cy

cz

cx

/0

cy

ly

lz cz1 0

1/

cx

lx

s

a

2 3llie

llcl

c sgsCf 444 5 ~2678 (5)

where ls9 and ls99 are the first and second time derivative ofls , which is the local level frame representation of the

vector going from origin of the local level frame to that ofthe cube frame (fig. 1).

The corrected IMU gyroscope output is a measure of theinertially referenced angular velocity of the cube frame incube frame coordinates . This angular velocity may bewritten as the sum of the cube frame representation of theearth’s angular velocity c

ie: and that of the angular velocity

of the cube frame relative to the local level frame :

clc

cie

cic

::: 68 (Roth, 1999). (6)

The resolution of commercially available IMUs variessubstantiall y. Therefore with regard to the theory, it isuseful to distinguish between high-grade IMUs which canresolve the Coriolis acceleration, i.e. , for an antennaspeed of motion as low as 2 km/h and the earth’s

angular velocity , and low-grade IMUs which cannot.Accordingly, when working with a low-grade IMU themeasured specific force cf can be approximated as

; <llcl

c gsCf 7= >> (7)

and the measured angular velocity as

clc

cic ?? = . (8)

The IMU that we have tested is the DMU-VG (CrossbowTechnology, 1997). Having an accelerometer resolution of0.012 m/ 2s and a gyroscope resolution of 0.03 deg/s, it canbe classified as a low-grade IMU. A high-grade IMU is forexample the TGAC-RQ (iMAR) with an accelerometerresolution as low as 510@ m/ 2s and a gyroscope resolutionof 0.0003 deg/s. This level of resolution of course alsomanifests itself in the TGAC-RQ’ s price, which lies in theUS$ 55,000 range in contrast to US$ 4,000 for theDMU-VG.

Initial Alignment

Before the start of motion, the initial orientation of theantenna needs to be determined. This is achieved by the so-called initial alignment.

The initial alignment of strapdown inertial navigationsystems used in aircraft and missiles is usually fullyanalytical, meaning that the full initial orientation (tilt andazimuth) is determined computationally from the IMUoutput. A fully analytical initial alignment requiresknowledge of the directions of the gravity vector g and the

earth’s angular velocity ieA in the navigation frame, in this

case the local level frame, and measurement of their cubeframe components (Britting, 1971; Chatfield, 1997).Therefore a fully analytical initial alignment is on principleonly possible with a high-grade IMU. However with regardto GPR antenna navigation, even with a high-grade IMU aproblem arises from the fact that the direction of ie

A in thelocal level frame is not known a priori. This problem doesnot exist for aircraft and missile navigation since thenavigation frames used in these applications are welldefined with respect to a geodetic reference earth model(Chatfield, 1997). These navigation frames however havethe disadvantage that they are not well defined with respectto the GPR survey area.

These problems can be overcome by an initial alignmentprocedure which is partly physical and partly analytical(Roth, 1999). First, the antenna is rotated until it is alignedwith the x axis of the local level frame (denoted as lx infig. 1). As a result of this rotation the yaw angle, describingthe azimuthal antenna orientation, becomes zero. Second,the tilt of the antenna system, as described by the pitch androll angles, is determined analytically from the IMUaccelerometer output using the knowledge that the gravityvector g runs parallel to the z axis of the local level frame(denoted as lz in fig. 1). From the accelerometer output anestimate of the gravity magnitude can be calculated as wellwhich is used to correct the accelerometer output for gravityonce the antenna is in motion.

It should be noted that when working with a high-gradeIMU, the orientation obtained by the above procedure canbe used to calculate an estimate of the earth’s angularvelocity in local level frame coordinates from the IMUgyroscope output. Therefore subsequent fully analyticalinitial alignments become possible. The so-obtainedestimate of is also used in the attitude and the velocityupdating as explained below.

Attitude, Velocity and Position Updating

Once the antenna is in motion, the direction cosine matrixand the attitude quaternion lcQ change according to

equations (3) and (4). An updating algorithm for either thedirection cosine matrix or the attitude quaternion can beconstructed by solving these differential equations over theupdate time interval, the boundary condition being given bythe respective attitude parameter at the beginning of theinterval. The basic approach to this problem is to assume ananalytical form for the input angular velocity in the updateinterval, e.g. a polynomial in time, and then use eitherRunge-Kutta techniques or a Taylor series expansion up to

cic

Aclc

A

llies

>? ~2B|| ls>cie?

cic?

lie?

lie?

lcC

some order to find an estimate of the respective attitudeparameter at the end of the update interval. Given a low-grade IMU, the input angular velocity is the measuredangular velocity of the cube frame relative to the local levelframe . As for high-grade IMUs, the input angularvelocity is the measured angular velocity of the cube framerelative to the inertial frame c

icC . It should be noted that the

earth’s angular velocity in local level frame coordinates, which enters the high-grade updating as well , is

constant.

Another common approach in the design of attitudeupdating algorithms is to use the rotation vector concept(Roth, 1999; Savage, 1998).

Equations (5) and (7) define the primary operationsperformed to update the velocity lsD . First, the measuredspecific force cf is transformed to the local level frameusing the orientation information from the attitude updating.Second, the transformed specific force lf is corrected forgravity and, when working with a high-accuracy IMU, forthe Coriolis acceleration to obtain an estimate of theacceleration lsDD . This acceleration estimate is thenintegrated over the update time interval to obtain a newvalue for the velocity lsD . The gravity correction makes useof the gravity estimate obtained as part of the initialalignment. Due to the “short distance” nature of the antennanavigation problem, the gravity can be assumed constantover the GPR survey area. The Coriolis acceleration can becalculated from the earth’s angular velocity obtainedfrom the initial alignment and an approximation for thevelocity lsD based on extrapolation from past values. Theinitial value for the velocity lsD to be used in the updating iszero.

The antenna position may be defined by means of a vectora, going from the origin of the local level frame to themeasuring point of the antenna. As shown in figure 1, thevector a can be expressed as the sum of the vector s andanother vector d going from the origin of the cube frame tothe measuring point of the antenna, i.e.

clc

ll dCsa EF . (9)

Thus, the antenna position la can be calculated given ls ,cd and knowledge of the antenna orientation, as expressed

by the direction cosine matrix . The components of thevector cd must be measured after attaching the IMU to theantenna. The position ls is obviously obtained fromintegrating the velocity lsD . The initial value for the position

ls to be used for this integration can be determined fromthe initial antenna position (fig. 1), which must beknown a priori, cd and the initial antenna orientation.

NAVIGATION EXPERIMENTS

The testing of the DMU-VG consisted of two parts. First,the abili ty to determine antenna orientation wasinvestigated. The second part of the testing aimed atassessing the positioning accuracy that can be obtained.

For the purpose of the experiments, a navigation softwarepackage was developed, which integrates data acquisition,processing and display. The software neglects axismisalignments and accelerometer biases since these had notbeen specified by the manufacturer of the DMU-VG. Thegyroscope biases are estimated as part of the initialalignment. For the attitude updating, the software uses anattitude quaternion algorithm.

To evaluate the abili ty to determine orientation, differentrotational motions were applied to the DMU-VG by handwhich ended in the same orientation as at the start ofmotion. The estimated final orientations were compared tothe initial orientations obtained from the initial alignment.The results demonstrate that over periods of 60 seconds theaccumulation of orientation error is generally below 1degree.

For the position estimation testing, the DMU-VG wasmoved by hand along different known straight andcurvilinear trajectories on a horizontal tabletop (fig. 2).Position errors of up to 1.35 meters (240% over the distancetraveled) after only 7.73 seconds of motion were observed.This shows that the DMU-VG is far from being accurateenough to be used for GPR antenna positioning without theaid of external position or velocity information.

Figure 2: Experimental setup for the position estimation testing

clc

G

lie

H

lie

G

lcC

l0a

Laptop

DMU-VGwith power supply

DISCUSSION

Strapdown inertial navigation in principal offers manyadvantages over conventional GPR antenna positioningtechniques by its abili ty to provide real-time three-dimensional position and orientation information. Inpractice however, the error accumulation characteristic ofinertial navigation systems can be a problem, especiallywhen working with a low-grade IMU, as shown by theresults of the navigation experiments. The accumulation oforientation and position errors can be attributed to thepresence of scale factor errors, axis misalignments, inertialsensor biases as well as inertial sensor drift (bias drift).Position errors accumulate much faster than orientationerrors as a result of the double integration.

For accurate positioning, the precise calibration for scalefactors, biases and axis misalignment models isindispensable (Stave, 1996). In addition, it is important toknow the effect of the inertial sensor drifts alone, since theyaccount for most of the error accumulation of a fullycalibrated IMU. A way to measure this effect is to simplyleave the IMU at rest after the initial alignment andcalculate positions as if it was moving (Roth, 1999). In thiscase position errors are solely induced by inertial sensordrift. Doing so for the DMU-VG produced position errors ofthe order of a couple of decimeters after only 8.5 seconds.Hence calibrating the DMU-VG would still not make itsuitable for self-contained positioning. This is of course notthe case for high-grade IMUs, which use inertial sensorswith low drift properties such as ring laser gyroscopes(Lawrence, 1998). In fact, high-grade IMUs are generallyprecision calibrated by their manufacturers and the gyro andaccelerometer data are corrected for systematic errors beforethey are output.

In spite of the unsatisfactory performance of the DMU-VGwith respect to self-contained positioning, the achievedlevel of orientation accuracy should be sufficient enough tointegrate the DMU-VG with an odometer wheel. The wheelprovides information about distance traveled from whichposition can be estimated using the orientation informationof the DMU-VG. This configuration would extend theapplicabili ty of the odometer wheel to environments withtopography or where changing of the profile direction dueto obstacles is necessary. Frequent stops in which, apartfrom keeping the estimated yaw angle, the navigation isrestarted could be used to keep the accumulation oforientation errors at a minimum.

Another possible use of low-grade IMUs might be for moresophisticated “rubber sheeting” techniques which make useof the velocity information. The error growth in velocity isconsiderably less than in position since only a singleintegration is required for velocity estimation.

REFERENCES

Bochicchio, R., 1988, Computerized Geometric Correctionof Ground Penetrating Radar Images, MSc thesis,Colorado School of Mines, Golden, Colorado, USA,91p.

Britting, K. R., 1971, Inertial Navigation Systems Analysis,Wiley-Interscience, New York, 249p.

Chatfield, A. B., 1997, Fundamentals of High AccuracyInertial Navigation, American Institute of Aeronauticsand Astronautics, Reston, Virginia, 339p.

Crossbow, 1997, Product Information Sheet: DMU Six AxisDynamic Measurement Unit, Crossbow TechnologyInc., San Jose, California, USA, 5p.

Etkin, B., 1959, Dynamics of Flight: Stability and Control,John Wiley & Sons, New York, 519p.

iMAR, Product Information, internet sitehttp://www.imar-navigation.de

Kuipers J. B., 1999, Quaternions and Rotation Sequences,Princeton University Press, Princeton, New Jersey,371p.

Lawrence, A., 1998, Modern Inertial Technology, Springer-Verlag, New York, 278p.

Roth, F., 1999, Strapdown Inertial Navigation for GroundPenetrating Radar Data Acquisition: Theory andExperiments, MSc thesis, Dept. of Geophysics,Colorado School of Mines, Golden, 217p. + CD.

Savage, P.G., 1998, Strapdown Inertial NavigationIntegration Algorithm Design Part 1: AttitudeAlgorithms, Journal of Guidance, Control andDynamics, vol. 21, no.1, p. 19-28

Stave, L.P., 1996, Calibration of the DQI (Digital QuartzInertial Measurement Unit), Proceedings of the 1996Annual Meeting of the Institute of Navigation:Navigational Theory for the 3rd Mill ennium, June19-21, Cambridge, Massachusetts, USA, p. 607-615