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A Reduced-Order Model

Integrated GPS/INSfor

XiufengHe, Yongqi Chen and H.B. IzThe Hong Kong P olytechnic University

ABSTRACT

Th e dominant factor in determining thecompu tation time of the Kalman filter is the dimensionnof the model s tate vector. Th e number of computationspe r iteration ison the o rder ofn3. Any reduction in th enum ber of states will benefit directly in terms of increa sedcompu tation time.In this paper, a high ord er model inintegrated GP S/IN S is described first, then areduced-order model based on the high-order model, isdeveloped. Finally, a faster tracking approach for Kalman

filters is discussed.A typical aircraft trajectory is designed for acomplex high-dynamic aircraft flight experim ent.A MonteCarlo analysis shows that the reduced orde r modelprese nted in this pa pe r provides satisfactory accuracy foraircraft navigation.

INTRODUCTION

Kalm an filters have bee n widely used in theintegration of the GPSDNS system, both the dynamicmodel an d observation model are requiredto describe theintegrated system. Th e dynamic model is usually described

Authors Current Address:Department of Land Surveying and Geo-Informatics, The Hong Kong PolytechnicUniversity, Kowloon,Hong Kong.

Manuscript received May 1, 1997

0885-8985/98/ 10.00 0 1998 IEEE

by a linear differential eq ua tio n involving the system errof the INS and G PS systems. T he observation models arobtained from a combination of INS and GPSmeasurements.

on the dimensionn of the integ rated system model statevector. The nu mber of computations pe r recurrence isonthe or der ofn3. Any reduc tion in t he number of states wibenefit directly in computation time. Moreover, thereduction of an integra ted system mo del will bring manybenefits to enginee ring realization. Since most currentlyavailable contr ol design m ethod s only workon smalldimension systems; the com plexity of a higher order modoften m akes it difficult to ob tain a stable system.

Th e objective of model ord er reduction is to find alower order m odel which preserves the dynamicsof morecomplex high-order systems.In this pape r, a 21-stateshigh-order inte grated mod el is investigated first, basedonthis, a 15-states reduced-order modelis develop ed byeliminating states which are unobservable o r weaklyobservable.A Mon te Carlo analysis shows that thereduced-order model presented in this paper providessatisfactory accuracy for aircra ft navigation.

Th e computation timeof the K alman filter depend

DESCRIPTIONOF THE INTEGRATEDGPS/INS SYSTEM

The Dynam ic Error Model

as a local-level,NEU (no rth, east, and upper),wander-azimuth system.

GPS/INS can be written as:

In his paper, a n INS platform is considered

A high-order dynam ic error model of Integrated

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X ( t )= A ( t ) X ( t )+ G t ) W t ) 1)

where X(t) is derived of the state vector X(t), and X (t)consists of various e rrors such t hat

with

is attitud e (errorangles;is azimuth er ror angles;is err or in east longitude;is error in north latitude;is err or in altitude;is x-velocity error;is y-velocity error;is z-velociiy error;is wander-(pzimuth angle error;is x-gyro constant drift rate;

is y-gyro constant drift rate;is z-gyro constant drift rate;is x-gyro first-order M arov d rift;is y-gyro first-order Marov drift;is z-gyro first-order M arov drift;is x-accelerom eter zero bias;is y-accelerom eter z ero bias;is z-acce1e:rometer zer o bias;is clock bias a nd c lock drift rate, respectively.

Th e vector of dyn amics noise W (t) is given by

The matrix A(t) (21 by 21) is the integrate d systemdynamic matrix, it contains eighty-five perc ent ze roelemen ts. Th e matrix G (t ) (21 by 11) is the coefficientmatrix, it contains ninety-six perce nt zero ele ments. Th eyare sparse matrices.

The Observation Model

are the tightly coupled and the cascaded approach.Intightly coupled metho ds, satellite pseudoranges are usedas measurements by the integration filter. In the cascadedmethod , the measurem ent consists of the navigationsolution which is three-dimensiona l position an d velocity.Each of the m ethods ha ; its own advantages and

this example the tightly (coupledmethod is used.In theobservation m odel of inltegrated GPS/INS, themeasurem ent vectors consist of combinations of th e

Two traditional methods of GPS/INS integration

disadvantages. For a m ore detailed discussion see [ 5 ] . n

platform INS and GPS. Fro m th e platformINS position,the distance plj between the aircraf t and jth satellite can bcalculated. Let GPS pseudorang e m easuremen ts bePjithen the observation equation can be expressed as

Z,(O = PI, - P, (2)

where z pJ(t) is the discrepancy between the measurementgiven by the two systems. The n

I

P, =[b XJ2 +o/ - I2 +(z -Z\/)'l' + / (3)

where (x, y, z) is the tru e position of th e aircraft in theEar th Centered Ea rth Fixed (ECE F) coordinate system,(Xsj, Ysj, ZSJ is the jth satellite position in the E CE Fcoordinate system, 6, is the distanc e error, mainly causedby clock biases:

6pj = 6 , v j (4)

where 6iu is a clock bias, vj is the GP S receivermeasurement noise. Assume th at the platform INS

position ( Xi, yi, Zi) in the E C E F coordinate system, istransformed from a wander-azimuth system. Then

X I = x +sx,YI = Y +

I = z + , 5 )

wh ere (6x1,6y1, 6zl )is the positioningerror of the platformINS, Pij,is then w ritten as

P , KX/ --T, * b, Y\,)2 + Z/ -z,)21i ( 6 )Eq. (2) linearized form is:

sz t) = H ( t ) x ( t )+V t ) 7)where 62(4) is a correction to t he difference between theGPS and INS m easurement. H( t) is the integrated systemmeasure ment m atrix, it contains eighty perc ent zeroelements and is a spa rse matrix. V( t) is the GPS receivermeasurement noise.

MODEL ORDER REDUCTION METHODINVESTIGATION

Controllability and Observability

play a key role in the prese ntatio n a nd unde rstanding ofmode l reduction since the stability of the K alman filter isdete rmine d by these two basic characteristics.

The ideas of system controllability and observability

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T he con cepts of controllability and observability areembo died in the system controllability and observabilitygrammians. For system(1) and 7), the controllability andobservability gramm ians can be defined as[4]:

T he controllability grammian ism-

= l e A ' G GT e A T ' d t0

Th e observability grammian isrcJ

e = e A r ' H HT e A d t (9)0

Fo r system (1) and(7), if A(t) has a 1 of its eigenvalues inthe op en left half of the plane thenP and l s are the uniquesymmetric positive semi-definite matrices which satisfy thefollowing Lyapunov equation

+ H T H = OTh e connection of system controllability and

and 6 is said to b e [4]:a) f ; is positive defin ite if, and only if, (A,G)

b) 6 is positive de finite if, and on ly if, (A,H)

observability to

is completely controllab le;

is completely observable.Mathematically, t he co ntrollability and observability

of the system can be dealt with using the same metho d[l].He re we conside r observability. Th e observability of thesystem can usually be d etermined by examining theobservability matrix M [l]:

M = ...LH(n)A(n)A(nl). . .A(O)]

If M is a full-rank matrix, system(1) and (7) areobservable. Otherwise they a re unobservable.

and (7) because of the difficulty in determining the ran k ofmatrix M. Instead of determining the rank of matrix M,the observability of system(1) and (7) can be determined

It is not easy to exam ine ob servability in systems (1)

by computingthe singular valuesof the matrix M.

Conside r singular value decomposition(SVD) ofthe matrix M such that

M = U W T =

whereare t he singular values of the matrix M. If the nu mbe r ofnonzero singular values is equal to the dimensionn , thesystems (1) and (7) are observable; however, if the n um ber

UTU = I, VTV = I

of singular values is less than t he dimensio nn, the systemis unobservable.

(sparse), an approach called t he classical manne r ofexploiting sparsity can be employ edso as to carry out thesingular value decomposition of matrix M. A moredetailed de scription of this approac h is foun d in[7].

integrated systems is workab le theoretically, but n otsuggested in practice because of the com plex calculationA rough a pproa ch is used to ex amine the observability ofintegrated systems, th at is, studying th e possibility of

estimating the state from the outp ut. If a dynamic systemobservable, all the states of the dynamic equation can beestimated at the output. C ontrarily, if a state cannot beestimated from the output, it is unobservable.

Since matrices A( t) and H (t) have special properti

Th e above m ethod of examining the observability

Reduced Order Model for Integrated GPS/INSModel orde r reduction methods have been

presented in numerous research papers. Thes e methodscan be divided into two groups. The first group attem ptsretain the dominan t modes of th e original system, such aAggregation methods, Modal m ethods, Lyapunov functimethods and Perturbation metho ds [6]. Anoth er approais based on applying an identification proced ure to

input-outpu t d ata obtained by driving th e original systemwith a specific input. For inte grate dGPS/INS systems,since the methodsof model o rder reduction a re applied iengineering practice, we a re interested in the first groupmethods, tha t is, to eliminate som e states in order toreduce computational difficulties.

link between the physical system an d its m athematicalmodels. According to[2], those states which areunobservable contribute little to the response of thesystem.Thus, these states may b e e liminated while stillpreserving the basic performanceof the dynamics system.

For integrated systems(1) and 7), we can

determine which states are observable an d which areunobservable using a Kalman filter. Gy ros constant driftaccelerometers zero bias and user clock and clock driftbias states cannot be observed by th e K alman filter. Thusin system (l) , three states of gyros constant drift areremoved, and three states of accelerometers zero bias,which ar e considered white noise, a re eliminated. Sincethe use of the clock and clock drift bias states are requirein the tightly coupled case, they cannot be eliminated an

An approach to eliminating states is basedon the

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are treated as consider states increasing the filter'srobustness [ 5 ] .A redu ced-order dynam ic of the integratedGPSDNScan be w ritten ;as:

X/ ) k4, ,47Kp7sv,p, p,@7a&&E, 9 CW , , ,4, 3'

The matr ix A~( t ) (1 5y IS is the reduc ed-orderintegrated system dynamic matrix, the matrix G ~ ( t) ( 1 5y11) is the coefficient matrix. T he ob servation eq uatio n canbe written as:

where 6ZR(t)is a correction to th e difference betweenGPS and INS measurements, HR(t) is the reduced orderintegrated system measurement matrix, V(t) is themeasurement noise.

A FASTER TRACKING APPROACHFOR KALMAN FILTERJNG

Consider the discrete form of the above reducedorder integratedGPS/INSsystem as:

x(k 1) = q k . x ( k )+r ( k ) ~ ( k ) (15)

Z(k)= H(k) X + V ( k ) (16)

where Q, c (k) is a state transition matrix,y (k) is thecoefficient ma trix,H(k) lis the measurement matrix,cP(k),y(k) and H(k) are sparse m atrices.w(k) and v(k) are whitenoise with associated variance m atricesQ(k) and R(k),respectively.

A Ka lman filtering algorithm with closed loopcontrol is employed. Th e algorithm is summarised as

q k ik-1) = qk)p(k-i, k- +r(k)@)rfk)(17)

(18)

(19)

q k )= q k /k-l)HT(k)[l@jl'fk/k-l)@(k)+hfk)]-'

f l k k)= [I - K(k)H(k)llik k - 1)[1- qk)H(k)]'

IwwwSince the state vectors of integrated systems areerror elements, the conirol aim is to eliminate these e rrors

so a direct control law can be conside red as

U (k ) = -2 II k )State estimatevector can be expressed as:

i / k ) = K( k)Z(k ) (21)

IEEE AES Systems Magazine, March 1998

Table 1. Navigation Errors ComparingHigh-Order Model 21 States)

with Reduced-Order Model (15 States)

longitudeerror@)Height

i ( k / k - l ) = O 22)

where P(k/k) and P(k/k-1)are up date covariance matricesand red ict ed covariance matrice s, respectively.*(k / k)

stare vector, respectively.K(k) is gain matrix.

computation time concentrateson Q,(k)P(k k)@T(k)inEquation (17) [3]. Since state transition@(k)is a sparsematrix, tha t is, approximately seventy-five perce nt of th eelem ents of the matrix a re zero. If the multiplication withnon-zero elements need b e performed, this reduces thenum ber of m ultiplications. It is suggested individualelementsbe used for th e Kalman filter, instead of treatingmatrix @(k). Th us a large com utational savings can beobtained. Fo r@(k) P k/k) @ (k) calculations, whichrequire 2n3 = 2 x 153 = 6750 multiplications, onlyL= 55 - 8 = 2961 multiplications a re now require d, wherL and J den ote the number of non-zero elements and1elements in the state transition matrix, respectively.

and.Kk / k -1) are updated state vector and predicatedFo r the above K alman filter algorithm, most of the

2 2p

2 2

SIMULATION

T o investigate the behaviour of th ereduced-order integrated system as compared to the

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Integrated system

state

2 1-states

15-states

Table 3. Computation Time Comparisonof 15 States Reduced M odel with and without

Faster Tracking Kalman Filter

Computationtime

in each step

0.15 s

0.06 s

Faster Tracking

Approach for

Kalman Filter

No

Yes

high-order system, a Mo nte C arlo simulation test iscarried out.

An aircraft trajectory is designed for actual andcomplex high-dynamic aircraft flight. It is assumed that anaircraft takesoff, flies, and maneuvers for a total time5300seconds. Th e man euve rs include climbing, pitching,rolling, and turning.

erro r accuracy using high-order and reduced-orderintegrated models. Table2 shows computation timecomparing the high-order with reduced-order integratedmodel. Table 3 shows computation of reduced-integratedmodel with and without faster tracking approach.

Figures 1-3show navigation erro r curves ofhigh-order and reduced-order integratedGPSDNS,respectively.A compa rison of Figures1 and 2 show nodifference in position e rro r using high-order andreduced-order integrated models. The re is a smalldifference in attitude err or anglesx, y and velocity erro rdiv 8VxP,8Vxp,but the big difference in azimuth err or

angle 2 and z-velocity erro r occur using high-order a ndredu ced-o rder mode ls. This is caused by simplifying thegyro and accelerom eter models.

Table 1, previous pag e, shows aircraft navigation

Computation Time

in Each Step

0.06 s

0.02 s

CONCLUSION

Based on above sim ulation results, the followingconclusions can be draw n:

8 0 0E

a,2 600 .

0 2000 4000 6000

T i m s e c )

Pitch - - - - - - - R o l l - A z i m u t h.~

-._ ~

Fig. 1A. Aircraft Attitude Error C urves (2 1 States)

8U

Umu-2tal

U

s.

600

400

2 0 0

0

- 2 0 0U 0 2 0 0 0 4000 6000

T i m e s e c )

Fig. 1B. Aircraft Attitude Error Curves (15 States)

- 0.3Pk

- 0.1I -0 1

-0 3.-

-030 2 0 0 0 4 0 0 0 6

Fig. 2A. Aircraft Velocity Error C urves 21 States)

0.3L

P 0.15

-0.1.-8 -0.3->

-0.5 I I0 2000 4000 60Ti m e s e c

Fig. 2B. Aircraft Velocity Error Curv es (%States)

1) Reduced -order integrated models canprovide satisfactory accuracy for aircraftnavigation.

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20

n -200 2000 4000 6000

Ti m e s ec )x-ps i - - ~ - - - - y - p s i - - - -psi~ __

Fig. 3A Aircraft Position Error C urves 21 States)

20E

10ea 0

-10

L

CO

U).-

0 2 000 4000 6000

x-posi - ~ - - - y-posi - - - -posl.

Fig. 3B.Aircraft Po sition Error Curves 15 States)

2) The com putation time of the Kalman filteris propo rtionalto the cub e of the num ber ofintegrated system state vector. Th e timereduction was approximately60% by themodel reduction from21 Order to 15 Order,which brings benefit to en gineering

realization. The faster tracking app roachfor Kalman filters presente d in this pape ris efficient for the integra ted system, itreduce d comp utation time approx.67 .

3) Th e co nstant drift rate of gyrosis removedin reduce d-order integrated systems,infact, the c onsta nt drift rate of gyros canbe me asured and precompen sated inorder to further improve navigation

accuracy.

REFERENCES

[l ] Chen, C.T., 1984,Linear System Theory and D esign,

Holt, Rineh art and Winston.

[2]Davison, E.J., 1967,A Method for SimplifyingLinear Dynamic System,

IEEE Tran s. Autom atic Control, AC-12, pp. 119-121.

[3] Frank L. Lewis, 1986,Optimal Estimation- ith an Introdu ction to StochasticControl Theory,

[4] Mason, Jeffrey Eric, 1988,

Wiley-Interscience Publication.

Identification Using Low Order M odels, Ph.D thesis,University of California, Berkeley.

[5 ]Spiro P. Karatsinide s, Octobe r 1994,Enhan cing Filter Robustness in Cascaded GPS-INS Integrations

IEEE Transactionson Aerospace and Electronic Systems,Vol. 30, No. 4.

161 Zhou Tong, 1988,Improving Design Assessment and Simulationof Large-scaleDynamic Systems, Ph.D thesis,

Michigan S tate University.

171 Zah ari Zlatev, 1991,Computational M ethods for G eneral Sparse Matrices,

Kluwer Acad emic Publishers.

1998 IEEE Aerospace Conference

March 21-28,1998 Snowmass at Aspen, Colorado

Sponsor: Aerospace Electronics Systems Society of the IEEE

Contact:

Mike Johnson or Beth Leitereg2225 R o s c o m a re R o a d

Los Angeles , CA 90077-2222,USATelephone:310-472-8019

E-mai l : johnson@ee .uc la . edu Confe rence WWW address: www.aeroconf .org

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kalman filter for control of sattelites

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