Kinematic Gps_resolving Integer Ambiguities on the Fly

8/12/2019 Kinematic Gps_resolving Integer Ambiguities on the Fly

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KINEMATIC GPS: RESOLVING INTEGERAMBIGUITIES ON THE FLYPatrick Y.C. Hwang

Rockwell Interna t ional CorporationAvionics GroupCedar Rapids, IA 52498

ABSTRACTIn kinematic GPS, the ini t ial integer ambiguitymust be resolved ei ther by a stat ic survey over t ime,or by instant cal ibrat ion with a known basel ine oran antenna exchange . Whi le these s tandard methodsrequire maintaining a basel ine stat ionary to anearth-fixed reference frame during the ini t ial izat ion,ther e a re s i tua t ions when a t l eas t one of the rece iv-ers may be constant ly in motion. This paper pro-poses two ideas for adapting standard kinematictechniques to si tuat ions that do not natural ly al lowfo r the co nstraint of a f ixed baseline. The f irs t cal lsfor extract ing the information needed to resolve theinteger ambiguity from the very data col lected whilethe kinematic survey is in progress. The second ideaaddresses the use of the antenna exchange techniquefo r mobile platforms where the original locations ofthe antennas a re n ot l ike ly to remain s ta t ionaryduring the physical exchange. Both ideas count oninformat ion f rom addi t ional measu rements to aug-ment their respect ive measurement models.

INTRODUCTIONFor the past decade of GP S development, signif icantadvancements made in receiver and systems technol-ogies have improved th e pract icali ty and ap plicabil-i ty of ca rr ier phase methodologies. Today, thesemethodologies show potential usefulness beyondtheir surveying roots. Several key breakthroughswere especially significant in the evolution of thecontributing technologies, part icularly the conceptionof kinematic surveying [l].I n kinematic GPS surveying, i t is generally recog-nized that , regardless of how the survey is actual lyconducted, the ini t ial integer ambiguity must beresolved by choosing any one of three standardmethods depending on the circumstances con-s t ra in ing t he survey: 1)performing a stat ic survey;2) using the antenna exchange technique; or (3) cal-ibrat ing with a known basel ine. These methods al linvolve the special need to maintain two fixed pointsfor a baseline. However, if at least one of the receiv-

ers is located out at sea, in the air , or in space, thena basel ine that is stat ionary with respect to anearth-fixed frame of reference cannot be easi lyestabl ished.In exploring new concepts involving carrier phasemeasurements in mid-1989, the author came up withthe notion that , with redundancy from the use ofmeasurements f rom more tha n four sa te l li t es , a k in-emat ic survey need not s t a r t o ut wi th the twoantenna s be ing s ta t ionary . Al though th i s idea wasconceived of inde pende ntly, it was late r discoveredth a t t he same idea had, in fac t , a l ready been pro-posed by Peter Loomis of Trimble in a paper deliv-ered ear l i e r tha t year [2]. Nevertheless, there isenough of a difference in the two viewpoints to jus-t i fy a n independent re-proposi t ioning of this idea,which we shal l cal l kinematic-on-the-fly, a referenceadmit te dly more descript ive of th e problem th an ofth e solut ion. To complete the ini t ializat ion process,the Loomis viewpoint required what amounts to acomplete resolution of all the unknown variables. Onthe o ther hand, th e proposed model in th i s pa pertakes on a somewhat more optimist ic outlook in onlyrequiring a part ial resolut ion instead.A second idea put forth in this paper general izes akinematic-related concept commonly cal led theantenna exchange (or swap) for use on a movingplatform. The supplemental information needed tomeet the increase in complexity over the originalantenna exchange model is derived not from extrasa te l l i t es but ra ther wi th an ext ra antenna .To faci l i tate the main discussion later on, a tutorialsection follows where the analytical tools used inconjunction with explaining basic GPS surveyingconcepts will be introduced. This is, in turn, followedby formulation of the two ideas mentioned.

PERSPECTIVE ON CARRIER PHASEMEASUREMENT MODELSTo provide some background in rudimentary con-cepts of kinematic surveying, we sha l l look at var i-ous GPS carrier phase surveying models. Analysesof these models will concentrate on their solvability,

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whereby the concept of dilution of precision (DOP)will be used to determine their quality . The DOPfactor gives a simple interpretation of how muchone un i t of mea surement erro r con tr ibu tes to thederived solution for a given situation . Natu rally, thesmal ler the DOP, the bet ter the qual i ty o f the so lu -tion. For illust rative purposes, the exam ples adoptedare one-dimensional in position.S tat ic Survey ingIn the GPS sta t ic su rvey ing model shown in f ig -u r e 1, two stationary receivers located at the ends ofa baseline are requ ired to t rack and m easure thephase o f the GPS carr ier s ignal arr iv ing a t thei rrespect ive an tenna locat ions. I t i s p resumed t hat a l lchanges in the carrier phase are exactly accountedfor f rom the moment t rack ing beg ins when the f i rs tmeasurement is made.

- c$B = A$ = COS AXA@= A@o + AI#JN= COS 0 AXA = COS8 AX - A@N

w h ereA& = initial measurement;A@N= initial integer ambiguity (in number ofwhole cycles).

c lear f rom equat ion 1 that the two unknowns Axa n d A$N cannot be solved with one measurementequat ion at jus t a s ing le instan t in t ime. With a sec-ond measurem ent made p resumably a t a d i f fer-e n t t i m e tl, the m easurem ent s i tuat ion becomes two-dimensional in nature.

A solution exists for [Ax A& J] ~ if , and only if, the2x2-coefficient matrix containing the geometricparam eters is inver t ible . And if t hat i s the case, theDOP factor is used to provide an indication of thequal i ty of the so lu t ion . The DOP determines themagnification factor of the measurement noise thatis translated into the solution derived. These condi-tions constitute the central notion of observabilitythat is widely used in control and estimation theory.The DOP facto r fo r th is s i tuat ion is ob tained bygenerating the variance of the solution vector withrespect to a one-un it erro r var iance in the m easure-ment .

/

\ '\ , ' ..../A Separation Between BAntennas Ax

F ig u re 1. Static SurveyingAt the very first instance of achieving carrier track,the true relationship (i .e. the total phase delay)between the phase mea surements seen at bo threceiver locations is unknown due to the nature ofoscillatory signals. The unobservable portion of thisphase delay is, however, known to be some multipleof a whole cycle. This quantity representing thenumber of whole cycles needs to be found in theprob lem commonly referred to as th e ini ti l integerambiguityFor static surveying, the resolution of the initialinteger ambiguity has to be accomplished by observ-ing phase measurements over a period of time. It is

w h ere th e d e t e rm in an t D = cos O - cos Bo Note heretha t , being related to the inverse, the determ inan t i salso a good indicator of th e solution 's quality .From equat ion 3 the DOP factor associated withA@N s CO COS ^ + cos20,)/(cos 81 - cos O0 , which sug-gests two th ings: 1) he ra te of change of t he geom-etry fo r a f ixed t ime in terval determines theobservability of the measurement situation; and (2 )hypothetically, if GPS were a geosta tiona ry systemof satelli tes, cos in thi s problem would be a con-stant, and this problem would have no solution.In a real - l i fe three-d imensional G PS stat i c su rvey,the amount of time necessary to obtain a solution istypically on the order of 20 to 30 minutes fo r shor t( less than 1 km) baselines [l].The observation t imerequ ired increases wi th basel ine leng th as a resu l tof such unmodeled error contributions as atmo s-pheric delay decorrelation between the two paths,and d if ferences in the mul t ipath seen at the twoantenna locations.


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Kinemat ic SurveyingAs an extens ion to s ta t i c surveying, k inemat ic sur -veying a l lows a t l eas t one of th e anten nas to roamfrom point to point while keeping continuous trackof the carrier signal . Continuity in the carrier phaseprofi le measured provides the user with an exacthistory of posi t ion changes of the roving antennasince leaving i ts ini t ial locat ion. The resul tsobtained from this type of surveying operat ion canbe very accurate (sub-decimeter level) , provided thatsome sor t of ini t ial izat ion procedure to resolve theini t ial integer ambiguity had already been madebefore the roving antenna was moved. In f igure 2,the stat ic example of figure 1is extended to i l lus-trate the kinematic model . For the addit ional kine-matic motion, we use a new variable 6x to denotethe posi t ion increment from the ini t ial . By aug-ment ing equat ion 1 to include this motion, weobtain:

A 4 + 64= COS 0 * AX 6 ~ )A& + A& + 6+ = COS 0 * AX + COS 0 6~A& + 6c# = cos 0 6x + cos 0 * Ax - A&

(4 )

Position Initial SeparationIncrement 6x Between Antennas AX

Figure 2. Kinematic SurveyingAt this point , the variables Ax and AdN should havealready been solved from the ini t ial izat ion. Thisleaves 6x, which can be solved instantaneously fromthe s ingle measurement (A& + 64 . The implicat ionhere is that beyond the process of ini t ial izat ion, aslong as continuous carrier t rack ing is maintained,the posi t ion derived by kinematic means is obtainedin a s imi lar way and under the same observabil i tycondit ions as conventional posi t ioning usingpseudoranges . In par t , th i s means tha t the accuracyof a kinematic solut ion is dictated by the standardPosi t ion Dilut ion Of Precision (PDOP ) measu re usedfor GPS code posi t ioning. I t should be obvious thatthe same also applies to stat ic surveying when solv-ing for Ax af ter com plet ing i ts ini t ial ization ( i.e.w i t h A& solved).There are several ways to accomplish the ini t ial iza-t ion of a kinematic survey. The most obvious andperhaps least desirable is to run a t ime-consumings ta t i c survey. Having made one s ta t i c survey, the

sa me basel ine may , of course, be reused to ini t ial izeoth er kinematic surveys a s well, i f operat ional cir-cumstances permi t . We note here tha t knowledge ofth e base l ine antenna separa t ion impl ies knowledgeof the total phase delay, or equivalent ly, the ini t ialinteger ambiguity.The most in t r iguing a l t e rna t ive among in i ti a li za tiontechniques, though, is one cal led the antennaexchange (or swap) f i rst introduced by BenjaminRemondi of the National Geodetic Survey in 1985 [3].Th is technique ut il izes, as i t s principal idea, themovement of the antennas pioneered in the kine-matic approach to help speed up resolut ion of theini t ial integer ambiguity. I t suggests that by movingone anten na to the locat ion of the o ther , the to ta lphas e delay can be solved jus t as effect ively as wait-ing for the satel l i tes to move appreciably, as is thecase for a stat ic survey. However, since there is noway to phys ical ly merge two ante nnas t ha t need tobe occupying exact ly the same locat ion at the end ofthe antenna t ransfer , the next bes t th ing to dowould be to march the o ther antenna of f to thelocat ion vacated by the f i rst one. This completes theexchange I t can be seen from figure 3 t h a t t h eantenn a exchange can a lso be in terpre ted a s a k ine-matic-type movement of one antenna by the amount6x = -2Ax, while the other is kept stat ionary. As wasthe case in equation 1, we have two variables tosolve for ; therefore two me asurements a re needed.These measurements need to be made once eachbefore and af te r th e exchange of th e antennas .

Ax 6x = -2 AxFigure 3. Antenna Exchange - Two EquivalentRepresentat ions

Before exchange:A& = cos 0 * Ax - Al#JN

After exchange:A& + 64= COS 0 (AX + 6 ~ ) A&

= COS 0 * (AX - AX) - A&= - COS tJ * AX - A~$N

where 6x = -2Ax. The two simultaneous equationscan be ar ranged in to the fo llowing mat r ix equat ion:

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A solution for equation 5 exists by the simple argu-ment that i ts coefficient matrix is invert ible, pro-vided that cos 8 0. Here , the mat r ix de terminantis -2cos 8. In using cos 0 before and after , the equa-t ions suggest an instantaneous exchange, al thoughin real i ty, such a thin g would be an impossibi l ity.The point , though, remains t ha t th e in i t i a l in tegerambig uity can be resolved even if th at were th ecase. In other words, the ini t ial izat ion with theantenna exchange method is not dependent onchanges in satellite geometry. In fact, the solution ofequat ion 6 points out that the DOP associated withth e in i t i a l in teger A$N is totally independent ofgeometry.While the antenna exchange method provides thefastest way to resolve the ini t ial integer ambiguity,the two antennas must be close to one another atsome point of a kinematic survey for this to bepract i ca l . W here anten na separa t ions a re prohibi -t ively large, the conventional st at ic method is st i l lwidely resorted to.General ization to R eal-Life GP S ModelsTh e simple models used above to convey th e con-cepts of carrier phase surveying have also been use-ful in deriving insights to their l imitat ions. Int ransfer r ing these ins ight s to rea l -l i fe GPS s i tua-t ions, several issues must be taken into considera-tion. The discussion in this section covers the issuesof the benefi ts of higher samplin g rates, the eleva-t ion in spat ial dimensionali ty of the model andaccommodation of receiver t iming errors, and exten-sion of th e observabil i ty cri ter ia introduced with t heone-dimensional models to higher dimensions.In the analyses above where the change in geometryover a period of time is a key ingredient to the solu-t ion, we have kept the i l lustrat ive models to thebare essentials by using only measurements made attwo t ime instants. These two t ime points, therefore,represent the s ta r t and the end of a t ime in terva lduring which the rate of change in the satel l i tegeometry dictates the speed of th e solut ion's conver-gence. In an actual si tuat ion, though, measurementssampled in between the two boundaries of th i s t imeinterval also contribute useful information. Thus, upt o a point , the h igher the m easurement samp l ingrat e, th e faster the ini t ial ization can be resolved.However, when successive measurements becomecorrelated, such improvements with higher samplingrates gradually become negated.

In order to conver t f rom one to three spa t i a l d imen-sions, the num ber of posi t ion-related variables needonly be t rebled. With i t , the number of satel l i tesneeded to solve this expanded se t of v ariables willhave to be proport ionately increased. For the GPSsituat ion, there is also a t iming error in the receiv-ers that must be accounted for . It is seldom conve-nient or necessary to dea l with t he t iming er ror in acarrier phase surveying model only because thiserror is inest imable. To see this, we again resort toa simplified illustration. A th i rd var iable A t isintroduced into equation 1 o give

A& = COS 9 AX - A& + A t(7)

In order to solve this equation, we need more mea-surements. One way to sat isfy this may be to intro-duce a second satel l i te which then adds anotherini t ial integer variable. Another way is to add ath i rd measurement f rom t he or ig ina l lone sa te l l i teat a different time. In either case, or even whencombined, i t can be shown with the solvabil i ty anal-ysis used before that the sets of variables formu-lated simply cannot be resolved. Heuristically, thisis due to the inabil i ty of the measurement si tuat ionto dist inguish between the ini t ial integer ambiguity,A&, and th e t iming er ror , At, both of which residein the range (measurement) space of the model .Hence, to retain the form of the more insightfulmodels given by equations 2, 4, a nd 5 , we el iminatethe t iming error from considerat ion al together. Thiscan be real ized by fo rming th e so-cal led double dif-ference which is made from the single differencemeasu remen t, A4, between two satel l i tes. Assumingtha t the phase measurements a re made f rom bothsatel l i tes at the two antennas nearly simultaneously,the t iming error conveniently cancels out in thedouble difference measurement formed. In so doing,the ini t ial integer is now a difference between apa ir of integers each associated with a A$. Thenum ber of double difference measurem ents wil lalways turn out to be one less than the total num-ber of satel l i tes being tracked.Final ly, the use of th e coefficient matrix determ i-nant and the DOP fac tors as indicators of the solu-t ion's accuracy is also val id in three-dimensionalsi tuat ions. A DOP measu re of solut ion accuracy canbe derived for each of the different measurementsi tuat ions. As pointed out before, for kinematic posi-t ioning beyond ini t ial izat ion, i ts DOP measure isidentical to the PDOP from conventional GPS codepositioning.

KINEMATIC ON T HE FLYA kinematic survey has t radi t ional ly been made upof two parts , the ini t ial izat ion, and then the survey-

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ing. When dealing with post-processed data, theorder of the two is unimportant. Sometimes, forredundancy, the init ial ization procedure is even car-ried out twice, once before, and then once after thesurvey for verification of the init ial integer ambigu-ity. In realt ime, the init ial ization has to be per-formed and completed first in order for thesurveying portion that follows to be meaningful. Toreiterate, the choice of initialization methods variesf rom running a t ime-consuming static survey, toinstant techniques l ike calibration by using analready-surveyed baseline, or performing an antennaexchange. All of the se alterna tives ca rry the obviouslimitation of requiring a baseline that is fixed to theG PS reference f rame a t leas t fo r the durat ion of theinitialization.Nonstatic Init ial izationTo combine the two parts of the kinematic probleminto one, let us first bring back equation 4 describ-ing kinematic surveying,

except that, now, Ax andClearly, we have three variables which can be solvedby pil ing on more measurements, but not from thesame satell i te over t ime. Such a measurement si tua-tion would be unsolvable because every measurementadded at a d i f feren t t ime po in t a l so adds a dif feren t6x variable, so there will never be enough measure-ments to deal with the set of variables created.However, by adding another satell i te to the mea-surement set , there is enough additional informationto overcome the previously underdetermined condi-tion.

are yet unsolved.

The paren thet ical superscr ip t ass igned to a variabledenotes the index of the satell i te associated with i t .The determinant of the coefficient matrix in equa-t ion 8 is cosAs we might have expected intuit ively, the determi-nant is maximal if the temporal change betweencos 0, and cos 01 and th e spat ia l ( satel l i te) separa-tion between cos 0(li and cos i z l are large.Three-Dimensional ModelThe one-dimensional example of equation 8 serves toil lustrate the fundamental principles involved. Thiswill now be generalized to thre e dimensions, where

cos 01(2)- cos 0,(lI cos 0,(2'.

we have th ree in i t ia l posit ion and th ree incrementalposit ion variables to begin with. In addition, we alsohave to add one init ial integer variable for everysatell i te pair (double difference). In all , we need atot al of six satell i te pairs to provide twelve double-d i f ference measurem ents (a t two t ime po ints) tomatch the variables generated: 3 initial positions, 3incremental posit ions, and 6 init ial integers. Thism e a n s t h a t a total of seven satell i te s ar e required tosatisfy this model.Although the five- and six-satell i te models are alsosolvable, they have significantly poorer observabili tybecause they rely on redundancy in measurementsmade over more than two t ime po ints to make u pfo r the lack of satellites. And when a model relieso n measu remen ts mad e at more than two t imepoints, i t is essentially attempting to derive infor-mat ion no t jus t f rom t he f i rs t -o rder , bu t h igherorder changes in phase as well . Such changes arecomparatively slow to evolve. This is not unlikeusing fewer tha n four sa tel l i tes fo r a s ta t ic survey .It is noteworthy here that to obtain results compa-rab l e t o a static survey made with four satell i tes, akinematic-on-the-fly survey requires at least sevensatel l i tes .Ka l man F i l te rBased on the above foundations, a three-dimensionalGPS measurement model can be formalized for anysuitable estimator. In this paper, we choose the Kal-ma n f i l t e r as the estimator for analytical purposes.

wher e is the single difference mea surem entf rom satel l i te i between the two an tennas , and[h,(i'h,(i)h,'i ']T is the unit direction vector to satel-lite i. N denotes the init ial integer variable in placeof A used before, with a reversal in sign.The f i rs t th ree components of the s ta te vector areposit ions , and the rem ain ing are in i t ia l in tegers(equation 9). In the Kalm an f i l ter fo rmulat ion , th einit ial posit ion and incremental posit ion variablesfor one dimension, Ax and 6x used in equation 11,ar e combined in to one s ta te th at i s a llowed to ra n-dom walk. Hence, the random process model for theposit ion states are made up of three independentrandom walk components , whi le the remain ing s ixin teger s ta tes are t reated as random constants.

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The implementat ion of the standard Kalman fi l teralgori thms with the parameters defined above wass t ra ight forward [4]. A simulat ion program was wri t-ten to generate a typical satel l i te constel lation, a ndto compute the relevant unit direct ion vectors. Thiswa s done to analyze the convergence of t he f i l tercovariance over t ime. Each plot in f igure 4 repre-sents the square root of the variance associated withan ini t ial integer state. Recal l that each integersta te corresponds to a satel l i te pair coupled as aresult of double differencing. Results underequivalent condit ions for a 4 -satel li te sta t ic survey-ing case are given in figure 5.An exact comparisonbetween the two cases is meaningless because themeasurement s i tua t ions a re d i ffe rent . Fo r the cho-sen s i tua t ions , though, the s ta t i c resul t s appear toconverge faster than those for the 7-satel l i te kine-mat ic .

sv14-sv13sv13-sv12sv12-sv11

Fi gu r e 5 . Ini t ial Integer Resolut ion: Stat ic4Satellites)

The mult iple plots of f igure 4 exhibi t varying ratesof convergence, something which can be attributed

to differences in satel l i te geometry. Although someof the integer est im ates converge upon their solutionfas te r tha n others , the to ta l so lut ion, including theposi t ion est imates, is not complete unti l al l the inte-gers have been resolved. Fortunately, i t turns outthat we do not need a total solut ion to complete theinitialization. As a matter of fact, we only needthree out of the six, represented below in equation1 0 by the double difference paired indices (a,a'),(b,b'), (c,c'), in order to proceed with a reduced-ordermodel.

&,ial-& #,ia'i-N(a, a'i hxlai-h x(a'l h iai-h ( a ' ) hz(aJ-hz(a'JA$(bl-A$(b)- b,b'J = h,(b)Lhx(b')h Y YbJ-h (b'J h,(bi-hz(b')A $ ~ c l - A $ ( c ' l - N i c , c ' ~ h,(c)-h,(c') h Y Ycl-h ( c J hz(c)-hz(c'i ]I [ (10)

Note tha t th i s mean s as few as four sa te l l i t es a reneeded once the t ransi t io n is made. However, whenthe three double-difference satel l i te pairs are madeup independently of d ifferent satel l i tes, th ere wouldbe a ma ximum of six sate llite s involved.The error covariance matrix furnished by the Kal-man fi l ter computat ions gives an accurate accountof the stat ist ics associated with the est imates i f themodel accurately reflects the t rue si tuat ion. Thisinformation can be used to determine the conver-gence cri ter ia for terminat ing the ini t ial izat ion.Com paring ViewpointsThe Loomis formulat ion for th e kinematic-on-the-flyproblem also uses a Kalman fi l ter for an analyt icaldescription [2]. I t t r ea t s th e in i t i a l in tegers , however ,as single-differenced integers, each associated withone satel l i te in formin g a single-difference measure-ment be tween the two antennas . Due to theinsolvability of a GPS model which includes a tim-ing e rror , a s described previously in conjunctionwith equation 7, the single-differenced integers arealso not observable as a consequence; only double-differenced integers (pairs) are. As a resul t , theLoomis formulation had to depend on the covarianceof the posi t ion variables a s a meas ure of conver-gence, which by themselves, r epresen t the total con-vergence of the entire solution.A s pointed out previously, though, we only require apart ia l convergence of three integ er pairs to beginthe survey. When we take into considerat ion thegood as well as the poor geometries tha t we have tocontend with, clearly it will take longer to resolveal l six integer pairs than for a subset of the fastestthree. For this reason, the viewpoint presented inthis paper is somewhat more optimist ic by compari-son for obtaining a solution.

A N T E N N A E X C H A N G E O N AMOVING PLATFORMThe original antenna exchange technique cal ls for amutual exchange of antennas between their f ixed


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locat ions. Since the exchange is a physical one, thebasel ine used between the antennas needs to remainunchanged during the t ime of exchange. This is nota problem with surveys done on sol id ground. How-ever , if th e antenn as a re in cons tant mot ion, thereis no way to determine where the original locat ionsof the an tennas were a t the end of an exchange .Sti l l , i f the spat ial relat ionship between th e anten -na s dur ing th e exchange i s e i ther known or m ain-tained, i t is possible to expand the use of theantenna exchange technique to a mobile platform.Moving Platform Ini t ial izat ionThe ini t ial izat ion technique proposed here cal ls for ath i rd antenna to be inc luded in an ar ray . This a r raymu st remain r igid ly f ixed to a p la t form th a t ca n bemoving, such as a ship's deck. In the array, theantennas are col l inear, and spaced by known dis-tances. For convenience, the formulat ion pursuedhere wil l assume equal spacing between the anten-nas .Car r ie r phase measurements a re made a t a l l threeantennas. Two sets of double difference measure-ments can be made by forming two antenn a pa i r -ings ; the th i rd pa i r ing is redundant . Then, bymutual ly exchanging two of the antennas, whileleaving the third antenna f ixed, al l ini t ial integerambiguit ies involved are instant ly resolved (Figure6). To see this, let us again resort to the kind ofsolvabil i ty analysis made before. This t ime, a two-dimensional example is used instead.Th e set of variables now comprises two ini t ia l posi-tions, Ax and Ay, two incremental positions, dx and6y, and four ini t ial integer ambiguit ies, the resul tantcombinat ion of two antenna pairs and two satel l i tes.This measurement model is described below byequation 11.

, G GAnt. 2 Ant. 0 Ant. 1Ant. 1 After Ant. 0 AfterExchange) Exchange)

The superscr ip t s - and denote t ime ins tant s beforean d af te r the a ntenna exchange respec t ive ly . Thesubscripts "01" and "21" refer to the pair ingsbetween antennas 0 and 1, a nd a n t e nna s 2 a nd 1.Without wri t ing out the ent ire inverse of the coeffi-cient matrix, the part ial solut ion is given below byEqua t i on 12 for jus t the in i t i a l in tegers .

2/3 0 -113 0 113 0 113 0 Ab,, '0 2/3 0 -1/3 0 1/3 0 113 A& '

N L l i L i= [2/3 -2/3 1/3 1 /3 2/3 2/3 2/3 2 /3 ;;;;;;@21 2i-Al#lLlil'+Al+5*liL'+

( i 2 jAs in the one-dimensional case (see equation 61, t h einteger solut ion is total ly independent of satel l i tegeometry. Rather, the non-zero numbers in the l in-ear connection matrix reflect the proport ion of thespac ing among the three antennas in the a r ray .Three-Dimensional ModelTo extend the example above to a GPS model , weneed to include variables for one more dimension ofposi t ion. Correspondingly, for the measurement set ,two more satel l i tes are added for a to ta l of four .The four satel l i tes provide three double differencesfor each of the two anten na pa i r s . Thus , by makingmeasurements before and af te r the antennaexchange, we have a total of 12 m e a s u r e m e n tsneeded to solve the 12 variables: 3 ini t ial posi t ions, 3incremental posi t ions, and 6 ini t ial integers.K a l m a n F i l te rWhen genera t ing a measu rement model for a Kal-ma n f i l t e r th a t i s based on th e above formula tion ,we can again combine the ini t ial posi t ion and incre-menta l pos i t ion var iables , jus t as was done for thekinematic-on-the-fly si tuat ion. The f irst three com-ponents of the state vector are posi t ions which aremodeled as independent random walk , and theremainder a re in i t ia l in tegers t rea ted a s randomconstants.

F i gu r e 6. Moving-Pla t form Antenna Exchange Array585


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where is the single difference measurementfrom satel l i te i between the two antennas m and n,an d [h,( i )h,( i )h,( i l ]Ts the unit direct ion vector tosatel l i te i .During the exchange of the antennas, no measure-ments are processed by the f i l ter . Over this durat ionof t ime, the measurement si tuat ion is undefined,being covered neither by equation 13 nor equa-t ion 14.

CONCLUSIONSThis paper has presented ideas that may be usefulin extending kinematic techniques to applicat ionare as beyond terre str ial surveying. By exploi t ing th eredundancy avai lable in more tha n four satel l i temeasurements, the requirement for stat ic ini t ial iza-t ion in kinematic surveying can be discarded. Thisinitialization, which resolves the initial integerambiguity, can instead be performed while the rov-ing receiver is in motion. Although there are t imesavings tha t may be derived from this, the real ben-efit of such flexibility lies in its applicability tovehicles that are constant ly in motion.To fu rth er expand th e ut i li ty of kinematic tech-niques to such si tuat ions, an augmentat ion to theoriginal antenna exchange technique was proposedfor moving baselines. This technique uses a collinearthree-antenna ar ra y to in it i al i ze a kinematic surveyfrom any mobile platform, such as the deck of aship , or a moving terrain vehicle. A potential appli-cation for this initialization technique involves therelat ive posi tion determ ination of an element which,

original ly located on the mobile platform, is thendispatched from the platform. One example of thiselement may be a series of buoys with hydrophonesdeployed and towed by a seismic surveying ship atsea. Or, i t may also perhaps be a reference target ,dropped by parachute , for camera o r ienta t ion in aphotogrammetric survey where no such f ixed refer-ences on the ground are avai lable or are within thecamera's field of view.By extending th e applicabil ity of kinematic tech-niques to posi t ioning and navigat ional set t ings, theoveral l usefulness of GPS is furt her enhanced. Aswe near the dawn of ful l GPS operat ion, the gradualuncovering of the system's t rue potential appearsstill to be incomplete. Meanwhile, its progress con-t inues to astound.

ACKNOWLEDGEMENTSThe author would l ike to thank Dr. Peter Loomis ofTrimble Navigat ion for bringing to his at tent ion thepaper that original ly proposed the redundant mea-surement solut ion to the kinematic-on-the-fly prob-lem [2].

REFEREN CES1. Remondi, B. W., "Kinematic and Pseudo-Kine-matic GPS," Proceedings of the Satellite Divi-sion of the Institute of Navigation's FirstInternational Technical Meeting, ColoradoSprings, Colorado, September 21-23, 1988.

Loomis, P. V. W., "A Kinematic GPS Double Dif-ferencing Algorithm," Proceedings of the FifthInternat io nal Geodetic Symposium on Satel l i tePositioning, New Mexico State University, LasCruces, New Mexico, March 13-17, 1989.

2

3. Remondi, B. W., "Performing Centimeter-LevelSurveys in Seconds with GPS Carrier Phase: Ini-tial Results," Navigation Journal of the Inst i -tute of Navigation, Vol. 32, No. 4, Winter 1985-86.

4 Brown, R. G., Introduction to Random SignalAnalysis and Kalman Filtering Wiley, NewYork, 1983.

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Kinematic Gps_resolving Integer Ambiguities on the Fly