XVI CONGRESSO NAZIONALE AIDAA 4-8 settembre 00 PALERMO NEW ALGORIHM FOR AIUDE DEERMINAION USING GPS SIGNALS F. BERNELLI-ZAZZERA, M. MOLINA, M. VANOI, M. VASILE Dpartmento d Ingegnera Aerospazale, Poltecnco d Mlano, Mlano ABSRAC he paper presents methodology and results for an algorthm of nteger ambguty resoluton. he am s to valdate a new concept of GPS atttude sensor, wth every recever ndependent from one other. he algorthm uses an nstantaneous statc geometrc nequalty n order to reduce the nteger search. A batch-loss functon s evaluated for checkng the remanng ntegers and fndng the soluton. he pecularty of the algorthm s to fnd the rght soluton even for a coplanar antennas array. he expermental work s the proof of concept of the procedure for the Unversty Mcrosatellte PalaMede, for whch the foreseen baselnes are 40 cm. he procedure s demonstrated usng real data, collected from standard, not spacequalfed, GPS recevers. he synchronsaton between the recevers s reconstructed nterpolatng the acqured data.. INRODUCION he Global Postonng System (GPS) constellaton s wdely used for determnng the poston of a user near the surface of the Earth. Over the two last decades the technology has hghlghted the possblty of usng the GPS sgnals for space applcatons, such as on-orbt autonomous navgaton and atttude determnaton []. he key of the atttude measurement concept s the phase dfference between sgnals receved from dfferent antennas []. Snce carrer phase dfference measurements are ambguous due to the unknown number of nteger cycles between two sghtlnes, the nteger ambguty resoluton s a necessary task before determnng the atttude. hs work analyses the performance of the nteger ambguty resoluton algorthm for a coplanar baselne array and performs a frst atttude ntalsaton []. A moton-based algorthm has been mplemented. he advantages over other methods nclude (I) the possblty of solvng ambguty wthout atttude knowledge, (II) avodng large matrx nverson, (III) usng a non-teratve procedure. Frst the algorthm ntroduces a geometrc nequalty to reduce the nteger space search. hen a batch-type loss functon s used to select a canddate soluton. he fnal soluton s valdated by means of an ntegrty check on the error covarance matrx.
he system s verfed wth a GPS test faclty consstng of a telescopc bar mounted on the mock-up of the satellte, to allow easy baselne varaton. he bar atttude, and hence the satellte yaw, s estmated by the phase dfference observed from two offset antennas, lnked to a par of ndependent recevers. As shown n Fgure, the memorzaton of the raw-data on two dfferent computers permts to perform off-lne tests. baselne Master Slave PalaMede mock-up Fgure est faclty descrpton he synchronzaton of the two sgnals represents the crucal pont. he attenton wll be ponted on an nnovatve GPS sensor desgn. he recevers are completely ndependent; each one mantans ts own tme reference wth ts orgnal clock. he synchronsaton s reconstructed nterpolatng the data of the acqustons. hs step hghlghts the performance degradaton n the atttude estmaton accordng to the baselne length. he paper s organsed as follows. Secton presents the PalaMede msson. Secton descrbes the basc concept of the atttude determnaton usng GPS phase dfference sgnals. Secton develops the algorthm for the nteger ambguty resoluton for an array of coplanar baselnes. Secton 4 dscusses the procedure for the atttude ntalsaton. Secton 5 contans the valdaton of the algorthm presentng the performance of GPS commercal recevers and results of the tests. Fnally the paper summarses the research now n progress.. PALAMEDE MISSION he near-earth three-axs stablzed satellte PalaMede s a basc platform entrely developed by the students of the Aerospace Engneerng Department of Poltecnco d Mlano. he ams of ths project are testng and developng standard technologes, and to get the know-how for the bus-desgn of small and cheap mssons. he satellte wll be placed n a sun-synchronous polar orbt at an alttude of 800 km, and currently launch s scheduled for 00 [4]. he orbt selecton wll guarantee an nterestng coverage of the globe at low cost and provdes good lghtng condton both for the target and the orbt. he former s of prmary mportance for the cameras, whle the latter s related to the thermal and power subsystems. Power to all the subsystems wll be provded by panels mounted on the lateral faces of the spacecraft, provdng 40 W, and by a battery durng eclpses. PalaMede wll be Nadr pontng and equpped wth two CCD camera as payload. One colour camera drected towards the Earth surface. Appendages outsde man body (e.g. omn-drectonal antennas) wll guarantee downlnk communcaton. he ADCS ncludes three magneto-torquers and a momentum wheel for the control. Non strngent pontng requrements allow ntegratng three-axs magnetometer and coarse sun sensors for prmary atttude soluton. A system of four GPS antennas for testng a new approach to the problem wll be consdered as secondary payload and connected to a GPS
recever. For obvous vsblty reasons the GPS antennas wll be placed on the zenth face, toward the deep space, wth a typcal reference baselne of 40 cm.. GPS BACKGROUND he atttude of a spacecraft s the orentaton of a body fxed frame wth respect to an external frame. he rotaton between the two references s represented by the transformaton matrx A. Performng an atttude soluton from GPS carrer phase nterpherometry s essentally a threestep process: nteger ambguty resoluton, atttude ntalsaton, atttude determnaton. Most atttude algorthms determne the atttude comparng a measured vector n the body frame wth the true vector n the external frame. Cohen [] appled ths concept to the subcentmetre precson of the GPS carrer phase. Due to the great dstance between the GPS satelltes and the recevers, the ncomng wave front can be consdered planar. he sgnal travellng at speed of lght reaches the antennas of the baselne at slghtly dfferent tmes. he measured carrer phase dfference between the antennas s used to determne ther offset. Snce the rotaton about the unt lne-of-sght vector can not be determned by a sngle GPS sgnal, the determnaton of the complete atttude requres at least an array of two baselnes onboard and two GPS satelltes avalable. he GPS satelltes generate a carrer sgnal n L band; ts frequency s centred on 575.4 MHz, correspondng to a wavelength of 9.0 cm. As llustrated, the phase dfference ϕ s related to the range dfference r between two antennas. he projecton of the baselne b nto the lne of sght drecton to the GPS satellte and s obtaned by: ( n) r = b cosϑ = λ ϕ () where b s the baselne length, ϑ s the front angle between the baselne and the lne of sght unt vector, ϕ s the fractonal phase dfference (n L cycles), λ s the wavelength of the GPS sgnal and n s the nteger phase dfference. he fractonal sngle-phase dfference ϕ, between master and slave antennas, s the fundamental measure and s expressed by: ϕ = b As + n () where s j R s the normalzed lne of sght vector to the GPS spacecraft n the external reference frame, R s the baselne vector (n wavelengths) n the spacecraft body frame, b A frames also known as the drecton-cosne matrx. he complete observaton equaton of the phase dfference measurement model can be wrtten as: x R s the atttude orthogonal matrx representng the transformaton between the two ~ ϕ = b As j + n + w () j j j where the ndces refer to the th baselne and the j th sghtlnes, w j ndcates the zero mean Gaussan measurements error wth standard devaton ϖ=0.5/λ=0.06.
4. INEGER AMBIGUIY RESOLUION ALGORIHM he nteger ambguty exsts whenever a baselne s defned, hence for b > 0. Snce the recevers only measure the fractonal part of the carrer phase, the range dfference s ambguous untl the nteger ambguty resoluton s solved. he atttude-ndependent algorthm presented n ths work refers to the method descrbed by Crassds, Lghtsey and Markley []. It s based on geometrc nequalty constrants to reduce the nteger space search and on a batch type loss-functon to be mnmzed n order to determne the optmal soluton. he procedure works n a couple of ways. he algorthm can use three sghtlnes and then consder one baselne at a tme, or can use three baselnes and then consder one sghtlne at a tme. Snce n ths paper t s assumed that the baselne array s coplanar, due to PalaMede confguraton, a set of non-coplanar sghtlnes must be avalable. κ are the ntegers assocated wth each baselne, where κ s twce the number of the wavelength contaned n the baselne. For three sghtlnes, the search space would consst of κ possble nteger permutatons. Applyng the nequalty to the th baselne at the ntal epoch, the number of combnaton falls down to the order of κ. he reduced subset conssts of the ntegers that pass the followng geometrc nequalty: b ( s s ) + ( ~ ϕ n ) ( ~ ϕ n ) ( ~ ϕ n ) ( s s ) + ( ~ n ) (4) > b ϕ hs equaton reduces the search space, snce for each baselne only two sghtlnes are consdered smultaneously, nstead of three. For a set of three coplanar baselnes the problem s solved consderng three non-coplanar sghtlnes. hs approach requres that the selected GPS satelltes reman avalable for the entre collecton perod. he ntegers to-be-determned reman constant throughout the entre data span. he soluton s ncluded n the ntegers that passed the geometrc nequalty. he nteger search s based on an atttude-ndependent batchtype loss functon: L + = - = Γ + J( n ) S (k) (k) log ( k) σ (5) k σ - ( Φ (k) n ) b trace{ S (k)} - - { S (k)} + ( Φ(k) n ) Γ (k)s (k) Γ (k)( Φ(k) n ) σ = trace (6) Γ [ s ϖ s ϖ s ] = ϖ (7) n n = n (8) n Φ ~ Φ ~ = Φ ~ Φ (9)
S ss s s = ϖ + ϖ s + ϖ s (0) where S and Γ are tme dependent. It can be shown that the expresson of J nvolves a scalar check on the norm vector resduals. For fndng the soluton, that mnmzes Eq. (5), mnmal moton s requred. In order to ensure the canddate soluton s optmum, the estmate error covarance s ntroduced: P L 4 [ Φ (k) n ][ Φ (k) n ] k= (k) = () σ he selected ntegers-set s checked ndrectly through the nequalty on the dagonal elements of P. he algorthm has the 0.00 probablty of selectng a wrong soluton f three tmes the square root of every dagonal element of P s less then 0.5. he dea of Eq. (4) s to create a reduced set of ntegers, usng two sghtlnes at the frst nstant of the acquston. he nequalty s used strctly to reduce the search space. Wth fast processors t s possble to get a soluton wthout nvolvng the nequalty, checkng Eq. (5) for all possble ntegers. For short baselnes ths can be done very quckly. 5. AIUDE INIIALISAION Once the nteger ambguty s solved, the atttude can be determned. he computatonally effcent RIAD algorthm [5] provdes three-axs atttude determnaton from two vector observatons. Because of ts smplcty ths non optmal (.e. determnstc) method has become one of the most popular for determnng frst three-axs atttude nformaton. he am s to fnd the atttude drecton-cosne matrx A, whch satsfes the relaton: (...n) A Vˆ = Ŵ = () where Vˆ,..., Vˆ n s a set of unt vectors expressed n an external reference system toward n known drectons and Ŵ,..., Ŵn are the same n unt vectors expressed n the bodyobservaton system. Once two non-parallel reference unt vectors Vˆ, Vˆ and the correspondng Ŵ, Ŵ have been selected, the goal s fndng the atttude matrx A whch satsfes: A Vˆ = AVˆ = () Ŵ Ŵ From the equatons above, matrx A s over-determned. he frst step conssts of constructng two orthonormal trads by means of reference and body unt vectors. Accordng to the notaton used for the nteger ambguty resoluton the trads are gven by the followng equatons: ( Vˆ Vˆ ) Vˆ Vˆ ŝ = ( Vˆ ( Vˆ Vˆ ) Vˆ ŝ = (4) Vˆ ŝ = Vˆ
( Ŵ Ŵ ) Ŵ Ŵ rˆ = ( Ŵ ( Ŵ Ŵ ) Ŵ rˆ = rˆ (5) Ŵ = Ŵ here s a unque orthonormal drecton-cosne matrx A whch satsfes the relaton gven by: A = M obs M ref (6) where M ref and M obs are [x] matrces, each defned by the three unt coordnate vectors of the trad: M obs [ rˆ M rˆ Mrˆ ] M = [ ŝ Mŝ Mŝ ] = (7) ref Equatons (6) and (7) defne the soluton of the RIAD algorthm. Due to the orthogonalty of the drecton-cosne matrx A, a necessary and suffcent condton s that the atttude soluton gven by Eq. (6) satsfes the followng relaton: Vˆ Vˆ = (8) Ŵ Ŵ he algorthm also develops a smple analytcal expresson for the covarance matrx of the RIAD atttude soluton. he conventonal atttude covarance matrx s defned as the covarance matrx of the Euler parametersaton angles. hs s very heavy to calculate and sometmes less useful than the covarance matrx referred to a set of angles n the body fxed system. he procedure shows the formalsm for the body-referenced covarance matrx and the correlaton to the covarance matrx wth Euler angles. he error angle vector s a set of angles, transformng the real atttude nto the measured atttude, defned as: [ δϑ δϑ ] δϑ = (9) δϑ thus, assumng the error n the angle to be small, for frst order lnearsaton: δϑ δϑ A = δϑ δϑ < A > δϑ δϑ (0) where <A> ndcates the expected value. he covarance matrx assocated wth the atttude matrx A s: where the error on the atttude matrx can be expressed as: P =< δ AδA > () δ A = A < A > () It can be shown that the Cartesan atttude covarance matrx s gven by: ( ½trace(P) ) I P Pϑϑ = ()
P + = Pobs APref A (4) he desred expresson of the Cartesan atttude covarance matrx, n terms of the observaton vectors, s: [( σ σ ) Ŵ Ŵ + ( Ŵ Ŵ )( Ŵ Ŵ Ŵ )] P ϑϑ = σ I + σ + (5) Ŵ Ŵ Ŵ σ = + (6) σ V + σ W σ = σ V σ W whereσ X s the varance of a component of Xˆ along the component normal to < Xˆ >. It should be hghlghted that even f usually the covarance matrx s computed n terms of Euler angles, the expresson of the Cartesan body-referenced error covarance P ϑϑ s more nterestng, n terms of evaluatng the relablty of the three-axs atttude determnaton. 6. RESULS A test faclty has been set up to evaluate the algorthm performance. Dfferent lengths have been consdered n order to evaluate ts consstency relatng to multpath effects. he separaton between the antennas was respectvely 60 cm, 00 cm, and 40 cm (Fgure ). In the frst two cases, the antennas are mounted on external appendages, n the last one they are drectly placed on the mock-up of the satellte. Fgure Baselne he recevers used were GARMIN GPS 5 LP-Seres wth channels. he real radated GPS sgnals provded from the recevers have been recorded and used for off-lne smulatons. he raw data from the recevers are sampled at Hz. In Fgure the dspersons of the pont made by the recever for the master and slave antennas are presented. hat gves an ndcaton on the performance of the recever employed durng the smulatons. In order to run the algorthm, at least three sghtlnes are requred: ths condton s verfed by the GPS constellaton geometry, that ensures a mnmum of four GPS s n sght at any tme. Moreover test condtons requre that the same three GPS sghtlnes are vsble for the whole duraton of the acquston, untl the soluton s found. Durng the collecton perod the software should keep track of the cycle, n order to avod cycle slp. In ths manner the ntegers to be determned reman constant throughout the entre data span.
Fgure a) Master Antenna Dsperson b) Slave Antenna Dsperson For a set of coplanar baselnes, the algorthm consders three sghtlnes and a baselne at a tme. he nteger computaton for each baselne s completely ndependent from the others. able summarzes the fundamental characterstcs of each test. As seen, the geometrc nequalty represented by Equaton (4) s used to sgnfcantly reduce the search space. he table refers to a sngle sghtlne and shows the dmenson of the reduced ntegers subsets, obtaned applyng the nequalty to the sghtlnes s and s. hs space can be further reduced consderng also the thrd sghtlne s. Whle executng, the algorthm sums the loss functon n Equaton (5) over tme for a perod correspondng to L epochs. est Length [cm] Integer Range Full Search Combnatons: κ Reduced Subset % A 60-8 to 8 4096 096 48.8 B 00-5 to 5 000 460 54 C 40 - to 64 6 75 able Integer Ambguty Resoluton results assocated wth dfferent baselne lengths he procedure converges when three tmes the square root of every dagonal element of the covarance error matrx P n Equaton () s less than 0.5. he use of standard GPS recevers has hghlghted the lmtatons of the system. Snce the standard software provded wth the recever outputs the acquston tme wth a resoluton of 0-8 seconds, beng the perod of the sgnal =0-0 seconds, the synchronzaton between the two recevers s not feasble wth the requred precson. hs means that a loss n cycles acquston occurs. he consequence s that the actual recorded data can be used to run the geometrc nequalty to show that an effectve reducton of the search space can take place. On the other hand the nteger ambguty resoluton s so strongly affected by the not synchronous phase acquston, that the attaned results are not relable for practcal atttude determnaton. As shown n able, dfferent values of L have been consdered durng the test campagn. he second column descrbes the convergence step; the thrd column contans the number of CPU flops requred to run the algorthm.
L [sec] Convergence Step Flops 0 6 5 6.99 0.64 60 4.89 00 7.89 able Convergence performance 7. CONCLUSIONS hs work addressed the frst two steps of the atttude determnaton problem and atttude ntalsaton from vector observaton. he nteger ambguty resoluton algorthm ncludes the advantages of both nstantaneous and moton-based technque. Frst t uses an nstantaneous statc approach to reduce the search space. hen a canddate soluton has been found assocated wth the mnmum value of a batch-type loss functon. he result s wth an ntegrty check on the error covarance matrx. Several are the advantages of the algorthm. Frst, t works even for coplanar baselnes. Second, the procedure s atttude-ndependent. hrd, the algorthm avods large matrx nverson. Fnally, t uses a non-teratve procedure and provdes an expresson for the error covarance matrx. he procedure s strctly correlated to the relatve poston of the sghtlnes used; due to the geometrc constrants t s not always possble to fnd a soluton. Once the ntegers for three sghtlnes and two baselnes are solved, the atttude can be determned and used to fnd the ntegers assocated wth the remanng sghtlnes. est sessons have been performed on the real spacecraft model. he analyss has hghlghted a strong correlaton between the soluton and the clock synchronzaton. he number of flops has been shown, n order to evaluate the computatonal cost of the procedure. he results wll be the startng pont for the atttude determnaton algorthm appled to the coplanar baselne array [6]. he deal s to get relable atttude nformaton from a set of antennas lnked to only one recever by means of a dedcated spltter. hs step wll hghlght the performance degradaton of the system related to the baselne length, lmts n terms of sample frequency and synchronzaton problems. REFERENCES. Cohen, C. E., Atttude Determnaton, Global Postonng System: heory and Applcatons, Volume, edted by B. W. Parknson and J. J. Splker, Vol. 64, Progress n Astronautcs and Aeronautcs, AIAA, Reston, VA, 994.. Cohen, C. E., Atttude Determnaton Usng GPS, Ph.D. dssertaton, Dept. Of Aeronautcs and Astronautcs, Stanford Unv., Stanford, CA, 99.. Lghtsey, E. G., Crassds, J. L., Markley, F. L., Fast Integer Ambguty Resoluton for GPS Atttude Determnaton, Proceedngs of the AIAA Gudance, Navgaton and Control Conference, Vol., Portland, OR, 999, AIAA Paper #99-967, pp. 5-. 4. Bernell-Zazzera, F., Ercol Fnz, A., Romano, M., Vasle, M., Prelmnary Desgn of the Mcrosatellte PalaMede, Internatonal Astronautcal Federaton Specalsts Symposum on Novel Concepts for Smaller, Faster & Better Space Mssons, Redondo Beach, Calforna, 999.
5. Shuster, M. D., Oh S. D., hree Axs Atttude Determnaton from Vector Observatons, Journal of Gudance, Control and Dynamcs, Vol.4, No., 98, pp.70-77. 6. Crassds, J. L., Lghtsey, E. G., Markley, F. L., Effcent and Optmal Atttude Determnaton Usng Recursve Global Postonng System Sgnal Operatons, Journal of Gudance, Control and Dynamcs, Vol., No., 999, pp.9-0.