On-the-fly GPS-based atttude determnaton usng sngle- and double- dfferenced carrer phase measurements Y. LI, K ZHANG AND C. ROBERTS Dept of Geospatal Scence, RMIT Unversty, GPO Box 476V, Melbourne 3001 VIC, Australa Tel: +61-3-995-3351, Fax: +61-3-9663-517, Emal: yong.l@rmt.edu.au M. MURATA Dept of Aerospace Engneerng, Natonal Defense Academy of Japan, Kanagawa, Japan ABSTRACT Carrer phase measurements are prmary observatons for GPS atttude determnaton. Although the satellterelated errors can be vrtually elmnated by formng sngle dfferences, the baselne-related errors, such as lne bases, are stll present n the sngle-dfferenced carrer phase measurements. It s, therefore, dffcult to resolve the sngle-dfferenced nteger ambgutes due to the lne bases. By formng double dfferences, the lne bases of the sngle-dfferenced carrer phase measurements can be effectvely removed. However, the man dsadvantages of ths method le n the fact that the double-dfferenced measurements are mathematcally correlated and consequently the atttude obtaned from the double dfferences s nosy. Ths paper presents a new algorthm through whch both sngle and double dfferences are used smultaneously to resolve these problems n real-tme. The soluton of the nteger ambgutes can be obtaned by searchng for the most lkely grd pont n the atttude doman whch s ndependent of the correlaton wth the double dfferences. Next, the lne bases and correspondng sngle dfference nteger ambgutes can be resolved on-the-fly by usng the nosy atttude soluton obtaned from the prevous double dfference procedure. In addton, the relatonshp between the physcal sgnal path dfference and the lne bas s formed. A new method s also appled to derve the atttude angles through fndng the optmal soluton of the atttude matrx element. The proposed new procedure s valdated usng ground and flght tests. Results have demonstrated that the new algorthm s effectve and can satsfy the requrement of real-tme applcatons. KEYWORDS: GPS, atttude determnaton, lne bas, nteger ambguty resoluton 1. Introducton Recent research has demonstrated that the Global Postonng System (GPS) can play a key role n many applcatons, e.g. spacecraft atttude determnaton (AD) and navgaton, due to ts long-term stablty, low cost and low power consumpton (e.g. Fuller et al., 1997; Purvgrapong et al., 1999; Um and Lghtsey, 001; Rechert and Axelrad, 001; Zebart and 1
Cross, 003). Current GPS AD algorthms can generally be dvded nto three functonal modules, namely lne bas soluton, nteger ambguty resoluton (IAR) and atttude angular soluton (Trmble Ltd, 1996). Lne bases are manly caused by the dfferences n cable lengths between antennas and the recever (Cohen and Parknson, 199) or dfferent rado frequency (RF) front ends n the recever (Purvgrapong et al., 1999) or a combnaton of both. They are usually treated as constant varables and calbrated by a procedure pror to startup of a normal AD procedure n GPS atttude determnaton recevers, e.g. Trmble s TANS Vector (Trmble Ltd, 1996) and Space Systems/Loral s GPS Tensor (Fuller et al., 1997). Another method s that the lne bases are treated as components of the state vector of the system, and therefore, estmated along wth other state components (e.g. Ward and Axelrad,1996; Purvgrapong et al., 1999). A GPS recever can measure only the fractonal part of the carrer phase. The nteger number of wavelengths between antenna and satellte s unknown. Ths s the well-known nteger ambguty resoluton problem. Two approaches have been developed to resolve the nteger ambguty problem for GPS-based atttude determnaton. The technques are ether motonbased (e.g. Cohen, 1996; Crassds et al., 1999) or search-based (e.g. Qunn, 1993; Knght, 1994; Sutton, 1997). Moton-based methods need to collect data for a perod of tme durng whch obvous changes of a vsble GPS constellaton or the host platform rotaton have occurred. The search-based methods use only sngle epoch measurements to fnd the most lkely soluton and these therefore occasonally are prone to ncorrect solutons due to measurement nose. Two technques have evolved. In the frst technque, the search s carred out n a real number doman. The search space conssts of all possble grd ponts of search parameters. These parameters can be the elevaton and azmuth angles of a baselne (Caporal, 001; L et al., 001) or the atttude angles of the host platform (Zebart and Cross, 003). In the second technque, the search s restrcted to the nteger number doman. The search space conssts of all possble combnatons of canddates of nteger ambgutes (e.g. Qunn, 1993; Knght, 1994; Sutton, 1997). The algorthms for atttude angular soluton can also be roughly dvded nto the followng two categores: (a) pont estmaton algorthms (e.g. Cohen, 1996; Crassds and Markley, 1997; Bar-Itzhach et al., 1998; L et al., 00) and (b) stochastc flterng algorthms (e.g. Ward and Axelrad,1996; Chun and Park, 001; Choukroun, D., 00). There are two types of pont estmaton algorthms. The frst type of pont estmaton algorthm uses vectorzed observatons (Crassds and Markley, 1997; Bar-Itzhach et al., 1998) and can be consdered as a two-level optmal estmaton problem (L and Murata, 001) the least squares problem and Wahba s problem (Wahba, 1965). A number of algorthms for resolvng the Wahba s problem have been proposed (.e., Wertz, 1984; Mortar, 1998). The second type of pont estmaton algorthm deals wth the dfferenced carrer phase measurements drectly. It uses ether a non-lnear, least-square ft (NLLSFt) method (Cohen, 1996) or converts the problem equvalently nto Wahba s problem (Cohen and Parknson, 199). IAR s usually an ntalzaton process snce nteger ambgutes are constant (assumng no cycle slps) and they do not need to be resolved agan once they have been fxed. The IAR and atttude angular soluton are therefore usually treated as two stand-alone procedures and they have been nvestgated separately n the lterature (e.g., Cohen and Parknson, 199; Knght, 1994; Crassds and Markley, 1997; Bar-Itzhach et al., 1998; Crassds et al., 1999; L et al., 00).The lne bas soluton s coupled wth the sngle-dfferenced IAR problem snce the lne bases reman n the sngle-dfferenced measurements. Ths means that t s necessary
to carry out data pre-processng for the lne bas soluton when IAR s carred out n the sngle dfference doman (.e. Trmble Ltd., 1996). Although one can derve the soluton n the double dfference doman (.e. Zebart and Cross, 003), the soluton from the double dfferences s less accurate than the soluton from the sngle dfferences. Ths paper wll combne all three modules for atttude determnaton n a compact form by ncorporatng the sngle and double dfferences to resolve the above problems. The man contrbutons n ths paper are focused on the followng aspects. Frst, a new algorthm that ams to fnd the optmal atttude matrx element soluton (AMES) s used to derve the atttude angular soluton. The AMES algorthm can be easly mplemented and flexbly appled to an arbtrary confguraton of antenna arrays (L et al., 00). Second, the nteger ambguty soluton n the sngle-dfferenced doman s obtaned from the coarse atttude soluton derved from the double-dfferenced measurements. The IAR procedure s carred out to search the atttude canddates n the double dfference doman to avod the problem caused by the lne bases. Ths arrangement also avods the correlaton problem of the double dfferences (Hofmann-Wellenhof et al., 1997). Lastly, the relatonshp between the physcal sgnal path dfference and the lne bas s formulated. A new algorthm for the pont soluton of lne bas s presented. Ths paper wll frst outlne the proposed GPS AD procedure and descrbe ts man operatonal modules. Then, algorthms for the atttude soluton, the nteger ambguty resoluton and the lne bas estmaton wll be presented n detal. The results usng the proposed procedure wll be appled to the ground feld tests and the flght experments wll be analyzed.. GPS-Based Atttude Determnaton Procedure A GPS-based atttude determnaton system usually conssts of at least two RF ports. Each port receves the GPS sgnals from an ndependent antenna. One can use two or more ndependent GPS recevers wth L 1 carrer phase output capablty to construct an AD system (Caporal, 001; L et al, 001). Due to the dfferences between recever clocks, the betweenstaton sngle-dfferenced carrer phase (SDCP) s not applcable to the dervaton of the atttude soluton, and the between-staton between-antenna double-dfferenced carrer phase (DDCP) must be used n such systems. Most commercal products use a common reference clock to convert the receved GPS RF sgnals nto the ntermedate frequency (IF) sgnals. IF sgnals wll be then correlated to demodulate GPS data and generate observatons such as pseudorange, Doppler, carrer phase and sgnal-to-nose rato (SNR). One beneft of usng a common clock reference s that the clock error s the same for all RF sgnals ncludng carrer phase measurements, and thus t can be removed by formng sngle dfferences. Ths s crucal for dervng the atttude soluton from the SDCP (.e., the carrer phase dfference between the GPS sgnals receved by two antennas separated by a short baselne). Ths knd of measurement also reflects the proecton of the baselne vector onto the lne-of-sght (LOS) vector to a GPS satellte. The soluton derved from SDCP s more accurate than that derved from DDCP, snce SDCP s less nosy. The SDCP measurement equaton can be wrtten as: T T A b n ŝ (1) 3
where the subscrpt denotes the th GPS satellte and denotes the th baselne, and s SDCP (n metres) assocated wth the th satellte and the th baselne, ŝ s the th unt vector of LOS, b s the th baselne vector, A s the 3 by 3 atttude matrx, n s the sngle- s the dfferenced nteger ambguty assocated wth the th satellte and the th baselne, lne-bas on the th baselne, s the wavelength of the GPS L 1 carrer sgnal, and v s the measurement error of SDCP assocated wth the th satellte and the th baselne. The lne bases n equaton 1 can be cancelled out by formng the double dfference (). The DDCP measurement equaton can be wrtten as: T T A b n ŝ () where s DDCP (n meters), ŝ s the dfference of LOS vectors between the th satellte and the reference satellte, n s the double-dfferenced nteger ambguty, s the measurement nose of DDCP. Fgure 1. Flowchart of the proposed GPS atttude determnaton algorthm 4
The proposed procedure for GPS atttude determnaton s presented n Fgure 1. It uses both SDCP and DDCP data and provdes the capablty to estmate lne-bases n real-tme. The nput data ncludes those for the postonng soluton (termed as Nav data n Fgure 1) and those for the atttude determnaton (termed as Obs data n Fgure 1). The Nav processng block calculates poston, velocty soluton and outputs LOS to the AD procedure as well. The dfferencng operatons ncludng both sngle and double dfferences are carred out when the LOS and observaton data become avalable. The IAR s used to fx nteger ambgutes n the DDCP doman f nteger ambgutes are not fxed. If ntegers have been fxed the processng wll go to the Yes branch to check and repar cycle slps. The nteger soluton from the search procedure s further checked for acceptance or reecton. If the soluton s reected, the processng returns the start pont and wats for observaton and LOS data of the next epoch. Otherwse, f the soluton s accepted, the processng begns to calculate the atttude angles. The modules for the lne-bas soluton and IAR n the SDCP doman run once the coarse atttude soluton has been obtaned from DDCP. The lne-bases are no longer needed for calculaton once they are fltered to suffcent accuracy. The fnal atttude soluton s derved from the SDCP. Any antenna n the array of the system can be used as the master antenna. Only Nav data from the master antenna s necessary, although those from other slave antennas are helpful n the processng. The poston soluton derved from the C/A code s accurate enough for atttude determnaton snce actual poston knowledge used n the AD processng derves from LOS vectors. Ths s based on the fact that the orbtal heght of GPS satelltes s about 10 7 m and the overall C/A pseudorange error s less than 100 m (ths s true snce SA has been swtched off), ths ntroduces a relatve error to LOS vectors at the level of 510-6. Ths means only an error of about 510-3 mm s ntroduced to the SDCP for a 1 m baselne. However the normal nose level of L 1 SDCP s about several mllmeters (e.g. 5 mm n Cohen, 1996). Thus, the postonng error can be neglected n atttude determnaton. 3. Algorthms 3.1 AMES Algorthm By takng nto account the capablty of trackng maneuvers, a pont estmaton algorthm rather than a flterng algorthm s used snce a flterng mechansm may cause a delay n response to the maneuvers. The conventonal pont estmaton algorthm s the NLLSFt method whch uses an teratve procedure to obtan the potentally hghest level of accuracy that a pont estmaton algorthm can acheve (Cohen, 1996). However, the nherent dsadvantages of the NLLSFt method are that t needs a coarse atttude to ntalze ts tmeconsumng teraton process. For real-tme applcatons, a straghtforward procedure s more acceptable, even f t would lose some accuracy. Based on ths consderaton, the AMES algorthm s adopted n ths paper. Detals of the AMES algorthm can be found n L et al (00) and s summarzed hereafter. A ( 9 1) state vector a to express the atttude matrx A s ntroduced n equaton 3, T T T T a [ a1 a a3 ] (3) T where a ( 1,,3 ) s the th row of A. 5
The cost functon of the AD problem can then be wrtten as follows, n m J ( a) w ( n h a) (4) 1 1 where m s the number of baselnes, and n s the number of vsble satelltes, weghted coeffcent, and h s a ( 1 9) matrx w s the T T T h [ b sˆ b sˆ b sˆ ] (5) x y z where b x, b y and b z are the three components of vector b. By ntroducng the followng vectors n T 1 u ˆ ( w sˆ sˆ ) [ w sˆ ( n )] (6) 1 s where =1,,,m, n 1 s ws s the weghted coeffcent assocated wth the th satellte, and soluton of the th baselne n the reference coordnate system. The rows of A can be estmated separately as m w b 1 a ˆ ( d b )ˆ u, =1,,3 (7) T û s the where wbs the weghted coeffcent assocated wth the th baselne, and the vector of d s the th row of the followng ( 3 3) matrx T 1 D ( BW b B ) (8) where W b s a ( 3 3) dagonal matrx wth three dagonal elements of respectvely. B s 3m matrx whch conssts of m baselne vectors. w b ( =1,,3) The case that (BW b B T ) s sngular usually mples that the antenna array s of a coplanar confguraton. The weghted least squares soluton for coplanar baselne confguratons can be derved from equaton 7 as follows, a ˆ 6 1 m e 1 w b ( e 1 b x e b y )ˆ u, =1, (9a) By takng nto account the orthogonal property of the atttude matrx, the thrd row of A equals the cross product of the frst and second rows of A a 3 a1 a where (9b) m e 1 w b 11 b y (10a)
m e1 e1 wbbxby (10b) m e 1 1 w b b x (10c) e e (10d) 11e e1e1 Equatons 9a and 9b are further appled to the analyss of the expermental data. Note that the AMES gves the constrant-free soluton n whch the orthogonal constrant s not taken nto account. One can refer to the lterature (L et al., 001) for detals of a procedure to orthogonalze the AMES soluton. 3. Atttude-Based Search Method An effcent algorthm to fx the nteger ambgutes s crucal to the success of atttude determnaton. The search procedure heren s carred out n the DDCP doman snce the lne bases vansh n the DDCP. One can use the coarse atttude soluton from the search procedure to calculate the SDCP nteger ambgutes later. The search method used n ths paper s based on the ambguty resoluton functon (ARF) that can be parameterzed as angles of both elevaton and azmuth of a baselne vector (Caporal, 001; L et al., 001). ARFbased algorthms for IAR can also be found n other geodetc applcatons (.e., Han and Rzos, 1996). For a GPS-based atttude determnaton system wth a mult-baselne confguraton, more than one ARF functon has to be used to coordnate each baselne f ARF s parameterzed on the angles of elevaton and azmuth. A more effcent method for ths stuaton s to defne the ARF by usng the atttude angles of the host platform as the parameters ARF (, p, y) 1 n m n m 1 1 cos [ ˆ s T A T (, p, y) b ] (11) where the search parameters of, p and y represent three atttude angles of roll, ptch and yaw respectvely. For the convenence of computer programmng, the reference satellte n the DDCP s counted n the calculaton of equaton 11. Ths also avods sortng the array of the DDCP agan f the ndex of the reference satellte n the array changes. Fgure a s the ptch-yaw ARF mesh plot wth = 0, p [ 90,90], and y [ 180,180]. The soluton pont s at (, p, y) = (0,0,119). Fgure (b) depcts the contour plot wth gradent drectons (as the arrows ndcated n the fgure) wthn the area around the soluton pont that s p [ 30,30], y [ 90,150]. It s also shown n ths Fgure how the ponts converge to the correct soluton. The soluton les at the sharpest peak. 7
(a). Ptch-yaw ARF mesh plot ( = 0) (b). Ptch-yaw ARF contour plot ( = 0) Fgure. The D and 3D plots of the ptch-yaw ambguty resoluton functon The search procedure based on equaton 11 can be carred out by takng trals of all possble grd ponts for atttude angles. Wthout losng the generalzaton, suppose there are sx satelltes n vew and three baselnes n the structure. The redundancy of the atttude-based ARF search method s 1 (53-3) and the redundancy of the search based on the elevaton and azmuth s 3 (5-). Obvously, the atttude-based search procedure usually provdes greater redundancy than the elevaton-azmuth-based search procedure. Ths mples that the atttudebased search can gve a more relable soluton from a statstcal pont of vew. The effcency of the search procedure takes advantages of both the hgh accuracy of the baselne length and 8
the coarse atttude knowledge. However any carrer phase nose, multpath and other perturbatons wll potentally prevent the ARF search from achevng the correct soluton. Once the DDCP nteger ambgutes have been fxed, a coarse atttude soluton can be further derved from DDCP. Ths coarse soluton s usually accurate enough to derve the lne bases as well as to fx the SDCP nteger ambgutes. 3.3 Pont Soluton of the Lne Bas If the lne bas s treated only as a term n the mathematcs regardless of ts actual physcal meanng t does not affect the processng of the atttude determnaton. However, further understandng the lne bas physcally can be helpful n decdng the qualty of a soluton of the lne bases. Because the lne bases are actually the phase bases, whch reflect dfferent lengths of sgnal paths between antennas and the recever, the lne bas on the th baselne can be wrtten as f L1 ( ) c l l (1) where l s the dfference n length of the two sgnal paths. The master path s from the master antenna to the RF ntegrated crcuts (ICs), and the th slave path s from the th antenna to the RF ICs of the recever. f L1 s the frequency of L 1 carrer sgnal, and c s the lght speed n a vacuum. Equaton 1 reveals the relatonshp between the lne bas and the actual dfference of the sgnal paths. Furthermore, l can be treated as the dfference n length between the two RF cables whch connect the master antenna and the th slave antenna to the recever respectvely f neglectng the dfferences that exst nsde ICs of the recever. For example, = 3.806 cm, from equaton 1, l = / 6 mm. Ths mples the length dfference between the master cable and the th cable s about 6 mm. It s easy to understand now that the values of the lne bases change wth the envronmental temperature varaton. The man reason s the envronmental temperature varaton causes dfferent length varaton of the cables. If the cables were made from the same materals, the dfference would be very small as they share almost the same thermal characterstcs. Therefore real-tme lne bas estmaton capablty becomes especally mportant for an AD procedure n some applcatons,.e. space applcatons where temperature wll vary greatly and frequently when spacecrafts go out of, or fall nto, the shadow of the Earth. Unlke methods that treat the lne bases as state varables to be estmated along wth other atttuderelated unknowns, a new approach s a standalone process to calculate the lne bases as presented heren. From equaton 1, one can wrte down the lne-bas for each and as T T A b n ŝ (13) Even f n s unknown at ths step, one can stll estmate by calculatng ts sne and cosne values after convertng unts nto cycles nstead of meters as follows: 9
ˆ T T sn sn ( ˆ s A b ) (14a) ˆ T T cos cos ( ˆ s A b ) (14b) and then ˆ ˆ sn 1 tg ( ) cos ˆ, =1,, n. (15) There are n estmates of and the fnal soluton s gven as an average of n ˆ 1 ˆ (16) n 1 ˆ by Note that lne-bas values calculated by equaton 16 are wthn one cycle,.e., 0.5, 0. 5 0,. There s a useful rule. One can also choose another range that s to decde whch range of lne-bas should be adopted. Because cables are almost the same length n a GPS AD system and usually these cables are made of the same materal, the lnebases are always small and ther values are closer to zero than to one cycle. Thus, one can 0.5, 0. 5. More accurate lne choose the former range rather than the latter one,.e. bas soluton can be obtaned by flterng ths raw soluton. 4. Feld Tests A number of feld experments have been conducted to valdate the method proposed above. In the experments, the raw sngle dfference carrer phase measurements and LOS vectors were measured by a TANS Vector GPS recever, whch s a sold-state atttude-determnaton and poston locaton system wth a four-antenna array (Trmble Ltd., 1996). 10 Fgure 3. Trmble TANS Vector GPS recever wth four antennas
Fgure 3 shows a three-baselne confguraton n whch four antennas are arranged n a 41 cm by 41cm square platform. The baselnes can be expressed n the antenna coordnate system (or referred to the body frame system) as, b 1 l l 0 T, b 0 l 0 T, b 3 l l 0 T, where l = 9 cm. The frst experment was conducted at Beng Insttute of Control Engneerng on 3 December 1998. The recever was postoned wth a clear sky vew and consstently recorded (a) Lne bas soluton (b) Error of roll (c) Error of ptch (d) Error of yaw data for about one hour. Fgure 4. The lne bas soluton and the error of the atttude soluton n the feld test Fgure 4 shows the lne bas soluton versus GPS tme. Average lne bases are lsted n Table 1 where the results come from two experments that were carred out n the same condtons although dfferent seasons (wnter and summer respectvely). It llustrates that the temperature varances due to envronment show lttle dfference on the lne bases. Table 1. Comparson of the lne bas solutons n dfferent seasons (n cycles) Lne Bases 1 3 Wnter (3/1/1998) -0.00-0.083 0.064 Summer (3/07/1999) -0.149-0.071 0.074 Table. Evaluaton of atttude soluton error n the feld test (n degrees) Error of atttude soluton Roll Ptch Yaw Standard devaton 0.18 0.30 0.41 Mnmum -0.56-0.76 -.17 11
Maxmum 0.88 0.77 1.45 The average atttude angles are 0.18 n roll, -0.98 n ptch, and 38.77 n yaw respectvely. The error of the atttude soluton s lsted n Table and also shown n Fgure 4. Suppose that the measurement nose s at the level of 5 mm, an approxmate and general rule of thumb for atttude determnaton angular accuracy (n radans) for a representatve baselne length of L (n meters) s gven n (Cohen, 1996) as (n radans) 0.005/L (17) Ths would ntroduce an angular error of about 0.3 for a one-meter baselne. For the baselne confguraton n the experments, the longest baselne has a length of 0.58 m. Thus accordng to equaton 17, measurement nose would ntroduce an error of about 0.5 to the atttude soluton. From Table, t can be concluded that the AMES can acheve the nomnal level of accuracy. 5. Flght Tests The GPS recever used n the tests was developed n the Natonal Space Development Agency of Japan. The system has four RF ports and each port has eght channels. It can therefore track up to eght satelltes smultaneously. Current confguraton of the recever unt contans a clock as a common reference to tag the tme of the measurements of pseudorange, Doppler and carrer phase at a rate of 1 Hz (L et al., 003). A number of flght experments were conducted n November 001. The GPS atttude unt was mounted on the body of the Dorner-8, Natonal Aerospace Laboratory of Japan s arcraft, whch s shown n Fgure 5(a). The antennas were arranged to form an approxmate 850 mm by 900 mm coplanar square confguraton as shown n Fgure 5(b). Baselnes n the body frame are numercally defned as (n unt of meters), b 1 ={0,0.849,0}, b ={-0.9,0,0}, and b 3 ={-0.9,0.853,0}. (a) Dorner-8 arplane (b) Baselne confguraton Fgure 5. GPS antenna mountng and baselne confguraton n actual flght experments An IMU (nertal measurement unt) wth three fber optcal gyros (FOG) and accelerometers were mounted on the arcraft to provde atttude reference to evaluate the GPS soluton. The stablty of FOG s, X-axs gyro: 0.08 deg/hr; Y-axs gyro: 0.46 deg/hr; and Z-axs gyro: 0.08 deg/hr respectvely. 1
(a) 3D traectory (a) 3D traectory (b) On-the-fly lne bas soluton Fgure 6. The 3D traectory of the flght experment and lne bas soluton The results of all flghts demonstrate the effcency of the algorthms, and the results of #3 flght only are llustrated hereafter. The poston and velocty are derved from the C/A code pseudorange measurements. The 3D traectory s depcted n Fgure 6(a) and the on-the-fly lne bas soluton s depcted n Fgure 6(b). It llustrates that the performance of the algorthm s excellent n the experments even durng the maneuvers. The atttude soluton obtaned from the GPS SDCP measurements by the AMES algorthm s shown n Fgure 7. The dfferences between the AMES soluton and the IMU output are presented n Fgure 8. The average and standard devaton of the dscrepances between GPS and IMU are lsted n Table 3. Both Table 3 and Fgure 8 have shown that the AMES and IMU solutons are consstent. For the baselne confguraton n the experments, the longest baselne has a length of 1.4 m. Thus accordng to Eq. (17), measurement nose would ntroduce an error of about 0.4 to the atttude soluton. From Table 3, t can be concluded that the AMES can acheve the nomnal level of accuracy. Moreover, the results llustrate excellent performance of the AMES to track maneuvers and a fast computaton speed. Table 3. The average and standard devaton of the dscrepances between GPS and IMU solutons (unt: degrees) Error of angles Average Standard devaton Roll 0.01 0.16 Ptch -0.056 0.13 Yaw -0.054 0.3 The atttude-based ARF search method has successfully passed the flght tests for both on-lne ntalzaton and n-flght reset capablty. Note that dfferent search modes are consdered n the procedure to deal wth dfferent stuatons that may occur n operaton. These nclude ntalzaton, operaton reset, changes of satelltes used n calculaton and detecton of cycle slps. All exhbt very satsfactory performance. 13
(7a) Roll (8a) Error of Roll (7b) Ptch (8b) Error of Ptch (7c) Yaw Fgure 7. GPS atttude soluton (roll, ptch and yaw) vs. GPS tme (8c) Error of Yaw Fgure 8. Dfference between GPS atttude soluton and output of IMU 6. Conclusons Ths paper has presented a procedure for the GPS-based atttude determnaton n real-tme through whch both sngle- and double-dfferenced carrer phase measurements are used smultaneously. Ths method can overcome the problem ncurred when only one type of measurement s used. Ths method can therefore solate the lne-bas problem from IAR n SDCP doman and obtan a much more accurate soluton from SDCP nstead of the nosy soluton from DDCP. 14
The paper has presented a seres of new algorthms that resolve the man problems exstng n the feld of GPS atttude determnaton,.e., problems for IAR, atttude soluton and lne-bas soluton respectvely. The experments have demonstrated that the proposed procedure can provde a very relable and effcent soluton of the atttude of host platforms. The ARF s parameterzed n the atttude angles and the search space s ndependent of the DDCP. Therefore the search, based on ths ARF, can avod the correlaton problem whch normally exsts when searchng for nteger canddates. The relatonshp between the lne bas and the dfference of physcal sgnal paths has been formulated. An algorthm for the pont soluton of lne bases has been presented and the experments have demonstrated ts hgh effcency. Ths makes the confguraton of the proposed procedure more flexble,.e. the module for lne bas soluton s treated as a standalone functonal block so that a pror calbraton for lne bases s no longer requred. The AMES algorthm can acheve the nomnal accuracy that a GPS AD system can reach n the experments. Its advantages such as excellent performance durng maneuvers and fast computatonal speed were also demonstrated n the experments. All of these features of AMES as well as ts straghtforward procedure make t very sutable for real-tme applcatons. 7. ACKNOWLEDGEMENTS We would lke to acknowledge fnancal support from the Chnese government postdoctoral fellowshp and the Japanese Scence Technology Agency (STA) fellowshp. The Beng Insttute of Control Engneerng (BICE) s thanked for ther provson of the TANS Vector s data and the Natonal Aerospace Laboratory of Japan (NAL) and the Natonal Space Development Agency of Japan (NASDA) for ther permsson to use the flght data n ths paper. The authors acknowledge Mr. Baoxang Sun and Mr. Yun Gao at BICE, Mr. Yoshyuk Ishma at NASDA and Dr. Masatosh Hargae at NAL for ther help and cooperaton. References Bar-Itzhach, I. Y., Montgomery, P.Y., Garrck, J.C. (1998). Algorthms for Atttude Determnaton Usng the Global Postonng System, Journal of Gudance, Control and Dynamcs, 1(6), 846-851. Caporal, A. (001). Basc Drecton Sensng wth GPS, GPS World, 1 (3), 44-50. Choukroun, D. (00). A Novel Quaternon Kalman Flter Usng GPS Measurements, Proceedngs of ION GPS- 00 (pp.1117-118), Alexandra, VA: Insttute of Navgaton. Chun, C., Park, F.C. (001). Dynamcs-Based Atttude Determnaton Usng the Global Posstonng System, Journal of Gudance, Control, and Dynamcs, 4(3), 466-473. Cohen, C.E., Parknson, B.W. (199). Integer Ambguty Resoluton of the GPS Carrer for Spacecraft Atttude Determnaton, Advances n the Astronautcal Scences, Vol. 78, 891-118. 15
Cohen, C.E. (1996). Atttude Determnaton, In: Parknson, B.W., Splker, J.J. (eds) Global Postonng System, Theory and Applcatons, Vol. II, AIAA, Washngton, DC, 519-538. Crassds, J.L., Markley, F.L. (1997). New Algorthm for Atttude Determnaton Usng Global Postonng System, Journal of Gudance, Control, and Dynamcs, 0(5), 891-896. Crassds, J.L., Markley, F.L., and Lghtsey, E.G. (1999). Global Postonng System Integer Ambguty Resoluton Wthout Atttude Knowledge, Journal of Gudance, Control, and Dynamcs, (), 1-18. Fuller, R., Hong, D., Hur-Daz, S., Rodden, J., Tse, M. (1997). GPS Tensor TM : An Atttude and Orbt Determnaton System for Space, Proceedngs of ION GPS-97 (pp.99-311), Alexandra, VA: Insttute of Navgaton. Han, S., Rzos, C. (1996). Improvng the Computatonal Effcency of the Ambguty Functon Algorthm, Journal of Geodesy, 70: 330-341. Hofmann-Wellenhof, B., Lchtenegger, H. and Collns, J. (1997). Global Postonng System: Theory and Practce, Fourth, revsed edton, Sprnger Wen New York, 191-196. Knght, D. (1994). A New Method of Instantaneous Ambguty Resoluton, Proceedngs of ION GPS-94 (pp. 707-716), Alexandra, VA: The Insttute of Navgaton. L, Y., Murata, M.(001). A Two-Level Optmal Estmator for Atttude Determnaton Usng GPS Measurements, Preprnts of 15th IFAC Symposum on Automatc Control n Aerospace (pp. 35 40), September, Bologna/Forlì, Italy. L, Y., Nakama, A., Murata, M., Isobe, T. (001). Atttude Determnaton Usng Two GPS Recevers for Antenna Control, Proceedngs of the 45th Space Scences and Technology Conference (pp. 1173 1178), October, Hamamatsu, Japan. L, Y., Murata, M., Sun, B. (00). New Approach to Atttude Determnaton Usng GPS Carrer Phase Measurements, Journal of Gudance, Control and Dynamcs, 5(1), 130-136. L, Y., Murata, M., Ishma, Y. (003). Flght evaluaton of New Algorthms for GPS Atttude Determnaton, Proceedngs of SatNav 003, The 6th Internatonal Symposum on Satellte Navgaton Technology Includng Moble Postonng & Locaton Servces (Paper No. 58), July, Melbourne, Australa. Mortar, D. (1998). Euler-q Algorthm for Atttude Determnaton from Vector Observatons, Journal of Gudance, Control, and Dynamcs, 1(), 38-334. Purvgrapong, S., Hashda, Y., Unwn, M.J. (1999). GPS Atttude Determnaton for Mcrosatelltes,, Proceedngs of ION GPS-99 (pp.017-06), Alexandra, VA: Insttute of Navgaton. Qunn, P. G. (1993). Instantaneous GPS Atttude Determnaton, Proceedngs, Proceedngs of ION GPS-93 (pp. 603-615), Alexandra, VA: Insttute of Navgaton. Rechert, A.K., Axelrad, P. (001). Carrer-Phase Multpath Correctons for GPS Based Satellte Atttude Determnaton, Navgaton - Journal of The Insttute of Navgaton, 48(), 77-88. Trmble Navgaton Lmted (1996). TANS Vector Specfcaton and User's Manual, Software Verson.10. Um, J., Lghtsey, E.G. (001). Atttude Determnaton for SOAR Experment, Navgaton - Journal of The Insttute of Navgaton, 48(3): 181-194. Sutton, E.(1997). Optmal Search Space Identfcaton for Instantaneous Integer Cycle Ambguty Resoluton, Proceedngs of ION GPS-97 (pp. 313-3), Alexandra, VA: Insttute of Navgaton. 16
Ward, L.M., Axelrad, P. (1996). A Combnaton Flter for GPS-Based Atttude and Baselne Estmaton, Proceedngs of ION GPS-96 (pp.1047-1061), Alexandra, VA: Insttute of Navgaton. Wahba, G. (1965). A Least Squares Estmate of Satellte Atttude, SIAM Revew, 7(3), 409. Wertz, J.R. (1984). Spacecraft Atttude Determnaton and Control D. Redel, Dordrecht, The Netherlands, 764. Zebart, M., Cross, P. (003). LEO GPS Atttude Determnaton Algorthm for a Mcro-satellte Usng Boomarm deployed Antennas, GPS Solutons, 6(4), 4-56. 17
Fgure 1. Flowchart of the proposed GPS atttude determnaton algorthm Fgure. The D and 3D plots of the ptch-yaw ambguty resoluton functon Fgure 3. Trmble TANS Vector GPS recever wth four antennas Fgure 4. The lne bas soluton and the error of the atttude soluton n the feld test Fgure 5. GPS antenna mountng and baselne confguraton n actual flght experments Fgure 6. The 3D traectory of the flght experment and baselne bas soluton Fgure 7. GPS atttude soluton (roll, ptch and yaw) vs. GPS tme Fgure 8. Dfference between GPS atttude soluton and output of IMU 18
Lne Bases 1 3 Wnter (3/1/1998) -0.00-0.083 0.064 Summer (3/07/1999) -0.149-0.071 0.074 Table 1. Comparson of the lne bas soluton n dfferent seasons (n cycles) Error of atttude soluton Roll Ptch Yaw Standard devaton 0.18 0.30 0.41 Mnmum -0.56-0.76 -.17 Maxmum 0.88 0.77 1.45 Table. Evaluaton of atttude soluton error n the feld test (n degrees) Error of angles Average Standard devaton Roll 0.01 0.16 Ptch -0.056 0.13 Yaw -0.054 0.3 Table 3. The average and standard devaton of the dscrepances between GPS and IMU solutons (unt: degrees) 19