AIAA--3947 ATTITUDE SENSING USING A GLOBAL-POSITIONING-SYSTEM ANTENNA ON A TURNTABLE by Mark L. Psak * Cornell Unversty, Ithaca, N.Y. 4853-75 Abstract A new atttude sensor s proposed that conssts of a sngle global postonng system (GPS) antenna mounted on a turntable wth ts phase center offset from the turntable's spn axs. It s beng consdered as a means of sensng 3-axs atttude nformaton. It s attractve because ts GPS recever could have a low number of channels and, therefore, be smaller and use less power. The system senses atttude by demodulaton of perodc oscllatons of the GPS carrer phase. These oscllatons are caused by the turntable s rotaton, and ther ampltude and phase depend on the drecton vector to the tracked GPS satellte. Ths system s descrbed n detal, ts demodulaton phase-locked loop s desgned, and ts performance s analyzed and evaluated va smulaton. The computer smulaton results show that, when usng a turntable radus of. m, a rotaton rate of 4, RPM, and an ovenzed crystal oscllator for the recever clock, the system can sense vector atttude wth a -σ accuracy of.4 deg at a bandwdth of.64 Hz. Accuracy can be mproved by ncreasng the turntable radus or by reducng mult-path reflectons. Introducton Many dfferent ar, space, and marne vehcles need a 3-axs atttude determnaton system, and varous types of sensor data can be used to determne the roll, ptch, and yaw orentatons. The measured carrer phase of a Global Postonng System (GPS) sgnal s one such data type. Atttude sensng based on GPS sgnals s attractve for several reasons. One s that a GPS recever often s already part of a system because of ts ablty to sense poston and velocty. If t can be made to sense atttude, then there wll be a weght and power savngs for the overall system because no addtonal atttude sensors wll be needed. Alternatvely, a GPS-based system can provde atttude determnaton redundancy. Yet a thrd attractve feature of GPS-based atttude sensng s the contnuous avalablty of ts sgnal. In low-earth-orbt spacecraft applcatons, atttude data from sun sensors and from horzon sensors may not be Assocate Professor, Sbley School of Mech. & Aero. Engr. Assocate Fellow, AIAA. Copyrght by Mark L. Psak. Publshed by the Amercan Insttute of Aeronautcs and Astronautcs, Inc., wth permsson. contnuously avalable, but a well desgned GPSbased system wll not have ths problem. The standard GPS-based atttude sensng method uses multple GPS antennas that are spatally dstrbuted on the user vehcle. The recever measures the carrer phase dfferences of the sgnal from a gven GPS satellte. Each phase dfference between an antenna par gves the cosne of the angle between the vector to the GPS satellte and the vector from one antenna to the other. The former vector s known n nertal coordnates; the latter vector s known n vehcle body coordnates. Gven enough of these cosne measurements, the full 3-axs atttude of the vehcle can be determned. The mnmum number of recever antennas s 3, and the mnmum number of receved GPS sgnals s. There are several dffcultes wth the standard approach. One s the need to resolve phase ambgutes, whch are nteger cycle uncertantes n the carrer phase dfferences. Another problem s that the recever may need to have many channels, one per antenna per satellte. A system that uses 4 antenna to track 6 GPS satelltes mght requre 4 channels. Ths can requre a hgh processor speed, whch can ncrease the recever's weght and power consumpton. Yet a thrd problem wth the tradtonal approach s that cosne-type atttude measurements are more dffcult to use n a full 3-axs soluton procedure than are vector-type measurements 3. Alternate schemes have been pursued for dong GPS-based atttude determnaton usng fewer than 3 antennas 4-6. The dea of Ref. 4 s to compute a pseudo-atttude based on the usual relatonshps between acceleraton, velocty, and atttude for an arcraft. References 5 and 6 use two GPS antennas that are mounted on a spnnng satellte. Ther approach makes use of the known dynamcs of a spnnng, nutatng spacecraft and deduces 3-axs atttude and atttude rate from the carrer phase dfferences between the sgnals that are receved at the two antennas. The present work presents a new way to use a sngle GPS antenna to sense 3 axs atttude nformaton. A patch-type antenna s mounted on a spnnng turntable. Its phase center s mounted off-axs, so that t translates around a crcle as the turntable rotates. Its feld of vew s centered on the turntable's rotaton axs so that ts geometrc gan pattern's nertal orentaton does not vary sgnfcantly wth table rotaton. Fgure
s a schematc depcton of ths system. Table rotaton axs and orentaton of antenna FOV center Turntable x ψ a GPS Satellte z Fg.. Measurement geometry for atttude sensng based on a GPS antenna mounted on a turntable. Ths system senses atttude by measurng a snusodal phase modulaton of the GPS carrer sgnal. The crcular moton of the antenna's phase center causes a receved GPS sgnal's carrer phase to have a snusodally varyng component because the moton creates snusodal varatons of the dstance from the antenna to the GPS satellte. Ths s effectvely an FM-type component. The frequency of ths modulaton equals the rotaton frequency of the table. The ampltude and phase of the modulaton can be deduced by a specally desgned phase-locked loop n the recever. Ths ampltude and phase are unquely related to the orentaton, n table coordnates, of the vector to the GPS satellte. Therefore, ths system provdes vector-type atttude measurements. A smlar concept has been used n the feld of rado drecton fndng 7. Ths system s related to the -antenna systems of Refs. 5 and 6. Those systems and the present system each make use of crcular moton of antenna phase centers n order to sense atttude. Atttude nformaton s derved from tme varatons of GPS carrer phase sgnals. There are some sgnfcant dfferences between the present system and those of Refs. 5 and 6. The present system does not rely on the spn of the vehcle to ω a r a θ s ρ s rˆs ψ s Patch Antenna y create antenna moton. References 5 and 6 depend on havng a good atttude dynamcs model of the vehcle, but the present system does not need any such model so long as the atttude varatons are not of too hgh a bandwdth. The systems of Refs. 5 and 6 use two antennas, but the present system uses only one. The present system can acheve a relatvely hgh-bandwdth, on the order of Hz. Atttude nformaton s derved wthn the phase-locked loop that the recever uses to track the GPS carrer sgnal. The systems of Ref. 5 and 6, on the other hand, have very low bandwdths Ref. 6 requres data batches of 5 to 4 sec n duraton n order to deduce atttude. Ths new system has been consdered because t has several mportant advantages. Frst, t can provde enough data to determne 3-axs atttude by trackng only GPS satelltes, whch requres only recever channels. Second, ths system does not requre resoluton of nteger phase cycle ambgutes because atttude s sensed from the tme hstory of the carrer phase of a sngle antenna, not from phase dfferences between multple antenna. Thrd, ths system can be bult mostly out of exstng hardware. The necessary hardware ncludes a recever whose phase-locked loops can be re-programmed and turntables of approprate dameter and speed. In fact, there exst open-archtecture GPS recevers whose trackng loops can be re-programmed 8, and there exst sutable turntables, ones whch are already beng used on spacecraft to provde atttude sensng whle smultaneously augmentng the ptch-axs angular momentum 9. Ths paper's 5 man sectons accomplsh ts goals of defnng, explanng, and evaluatng ths new system. Secton defnes the system's hardware confguraton. Secton 3 explans what ts measurements are and how these are related to atttude. Secton 4 desgns and analyzes a phase-locked loop that s used to demodulate the atttude nformaton. Secton 5 descrbes a smulaton that has been used to evaluate the system's performance. Smulaton results are presented n Secton 6 along wth analyss results. The paper closes wth a short conclusons secton that summarzes ts contrbutons. II. Descrpton of System Hardware Components Ths secton descrbes the basc hardware requrements for the desgn of the system that s depcted schematcally n Fg.. As stated above, the system needs a turntable, a GPS antenna mounted on the turntable, and a recever that s connected to that antenna. In addton, the turntable needs to have a speed controller, and t needs to have an encoder so that the turntable's rotatonal phase, ψ a on Fg., s
always avalable to the recever. The turntable s envsoned as beng lke a typcal spacecraft scan wheel that s used for smultaneous ptch axs momentum augmentaton and horzon sensng 9. Its dameter would be on the order of.5 m, and t would be able to rotate at speeds up to 4, RPM. Slower turntable speeds are acceptable, and larger turntable dameters wll tend to ncrease the system s accuracy, but these numbers have been used as baselnes because they are typcal of hardware that s currently used on a number of spacecraft. The accuracy of the turntable's poston encoder s mportant. The recever needs to know ψ a (t) n order to deduce the azmuth of the drecton vector to each tracked GPS satellte, ψ s. Any error n ψ a wll translate drectly nto an error n ψ s. Therefore, the requred encoder accuracy s. deg or better. The antenna should be a patch-type antenna. These can be made wth a dameter on the order of.5 m. Ths allows the phase center to be mounted at a sgnfcant dstance from the turntable rotaton axs. The nomnal mountng radus assumed for ths study s r a =. m. The antenna feld of vew must be farly wde, and ts center must be algned wth the turntable s rotaton axs. The dea s to have a wde enough feld of vew so that the system can always see at least GPS satelltes. System geometry s mportant n order to get good sgnal recepton. The turntable needs to be cantlevered on ts bearngs so that the antenna has a clear vew of the sky durng the whole rotaton cycle. The turntable should have a ground plane for the antenna;.e., ts outer face should be a ground plane. Also, the turntable/antenna system should be mounted on the user vehcle n a place that mnmzes mult-path nterference. It may be a good dea to restrct the antenna feld of vew somewhat f that wll help to reduce mult-path sgnal recepton. The recever can be a standard recever as n Ref. 8. There are three mportant features that the recever must have. Frst, t must be able to accept turntable azmuth readngs from the turntable encoder and synchronze them wth ts correlator accumulatons. Second, t must have a specal purpose phase-locked loop that measures the n-phase and quadrature components of the turntable-synchronous carrer phase oscllaton. Any open-archtecture recever should be modfable to have the requste phase-locked loop. Thrd, f one wants to do 3-axs atttude determnaton, then the recever must have at least channels so that t can smultaneously track at least GPS satelltes. The fnal requrement of the hardware desgn s that t must transmt the 575.4 MHz L sgnal from the antenna to the recever s rado-frequency (RF) front end wthout sgnfcant loss of sgnal-to-nose rato (SNR). In order to do ths, one must transmt the RF sgnal across a rotary jont, or one must mount the recever on the turntable. In the latter case, the system wll have to transmt power to the recever across the rotary jont, and the recever s atttude estmate wll have to get transmtted back across the rotary jont. If the recever s not mounted on the turntable, then a rotary capactve couplng can be used to transmt the RF sgnal across the rotary jont. III. Measurement Model The geometry of Fg. can be used to explan why ths system's measurements gve atttude. Assume that the rotatng turntable n the fgure s mounted on a user vehcle and that the user vehcle's atttude vares slowly compared to the rotaton speed of the turntable. The followng are the sgnfcant geometrc and knematc features of the system: The xyz coordnate-system s fxed to the user vehcle. It does not rotate wth the turntable, but ts z axs s algned wth the turntable's rotaton axs, and ts x-y plane s the plane n whch the patch antenna's phase center moves. The patch antenna's locaton n the xyz coordnate system s defned by the rotaton angle ψ a and the radal offset r a. The turntable rotates at a constant speed ω a. Therefore, ψ a (t) = ω a t + ψ a. The GPS satellte's poston n the xyz coordnate system s defned by ts azmuth, ψ s, elevaton, θ s, and dstance from the orgn, ρ s. Typcally, r a wll be on the order of. m, whle ρ s wll be on the order of 6 6 m. Therefore, (r a /ρ s ) <<. The user vehcle atttude can be determned f the recever can sense ψ s and θ s for two or more GPS spacecraft that are not collnear wth the user vehcle. The quanttes ψ s and θ s defne the drecton to the GPS spacecraft, rˆ s, n user vehcle coordnates. Gven knowledge of the user vehcle locaton, ths same vector s known n nertal coordnates. It s well known that one can unquely deduce 3-axs atttude gven knowledge of two or more ndependent drecton vectors both n vehcle coordnates and n nertal coordnates. Ths s why the proposed system can be used to determne the full 3-axs atttude f t can track or more GPS satelltes. In order to understand how to deduce ψ s and θ s from carrer phase measurements, consder the range from the user antenna to the GPS satellte. From geometry, the range between the antenna and the satellte s: ρ = ρ + r ρ r cosθ cos ω t + ψ ψ ) as s a s a s ( a a s s ra cosθs cos( ωat ψ a ψ s ρ - + ) () where the approxmaton on the second lne of eq. () 3
s vald for (r a /ρ s ) <<. The range to the GPS satellte can be used to deduce an expresson for the receved carrer phase. If ω c s the transmsson frequency of the sgnal n radans/sec, then ω c φ c (t) = ω c t - ρ as (t) + constant c = ω c t + φ Dopp (t) () where φ c s the receved carrer phase n radans and c s the speed of lght. The term φ Dopp (t) s the ntegrated effect on the carrer phase of the sgnal's Doppler shft. An alternate expresson for the receved carrer phase can be derved by substtutng the nd lne of eq. () nto eq. (): φ c (t) = ω c t + φ Dnr (t) + x c cos( ω at + ψ a) + x s sn( ω at + ψ a) (3) The term φ Dnr (t) s the Doppler-nduced phase perturbaton that would be present f there were no turntable rotaton. Ths quantty consttutes what s usually known as the ntegrated Doppler shft or the accumulated delta range. The last two terms on the rght-hand sde of eq. (3) gve the effects on carrer phase of the turntable's rotaton. The coeffcents x c and x s are ω x c = cra [ cosθ s cosψ s] (4a) c ω x s = cra [ cosθs snψ s] (4b) c The quanttes ψ s and θ s can be deduced from eqs. (4a) and (4b). Suppose that r a s known and that x c and x s have been measured by the recever Secton 4 of ths paper wll show how to do ths. Then the only unknowns n eqs. (4a) and (4b) are ψ s and θ s, and these equatons can be nverted to yeld the formulas ψ s = arctan(x s, x c ) (5a) θ s = arccos[( c x c + xs ) /( ωcra )] (5b) IV. A Phase-Locked Loop for Trackng Snusodal Carrer Phase Varatons A coherent GPS recever uses a phase-locked loop to reconstruct the carrer phase nsde of the recever. Fgure shows a hgh-level block dagram of a typcal channel of a coherent GPS recever, 3. The RF front end starts wth the sgnal from the antenna and preamp, y RF (t), and performs band-pass flterng and downconverson va mxng. Its output sgnal, y IF (t), has a nomnal ntermedate frequency (IF) of ω IF. The carrer phase numercally controlled oscllator (NCO) constructs n-phase and quadrature approxmatons of the down-converted carrer sgnal, cos[ω IF t + φ re (t)] and - sn[ω IF t + φ re (t)]. These sgnals are mxed wth y IF (t) to form the base-band n-phase and quadrature sgnals, y I (t) and y Q (t). The delay-locked loop (DLL) correlates these sgnals wth a reconstructon of the pseudorandom (PRN) code of the GPS satellte that s beng tracked, and t adjusts ts play-back rate of the PRN code so as to maxmze the correlaton. In the process, the DLL produces n-phase and quadrature accumulatons, I n and Q n, once every PRN code perod,.e., about once every. sec. The loop flter of the phase-locked loop (PLL) uses the I n and Q n accumulatons to adjust the frequency of the carrer phase NCO by adjustng ω re (= dφ re /dt). Ths quantty s nomnally the PLL's estmate of the carrer sgnal's Doppler shft because φ re (t) s nomnally the PLL's estmate of φ Dopp (t). In the present system, the PLL's loop flter s constructed to estmate x c and x s as part of the procedure by whch t computes ω re. Recall from eq. (3) that x c and x s are the coeffcents of the two components of the carrer phase that oscllate at the turntable frequency. The PLL estmates x c and x s as part of a Kalman flter. In order to develop ths Kalman flter, t s necessary to develop a stochastc model that descrbes the dynamcs of φ Dopp (t) and the effect of φ Dopp (t) on the measurements I n and Q n. Carrer Phase Model A dscrete-tme carrer phase model has been developed. Suppose that the DLL's PRN code cycles start and end at the sample tmes t, t, t,, t n, Then the carrer phase dynamc model s x p x v xa xc x s n = tn t n tn x p x v xa xc x s n t t n n 6 tn - ω re(n-) + wn (6a) φ Dopp (t n ) = φ re (t n ) + x p(n) + x c(n) cos( ω at n + ψ a) + x s(n) sn( ω at n + ψ a) (6b) In ths model t n- = t n - t n-. The frequency ω re(n-) s the value of dφ re /dt durng the tme nterval t n- to t n. The state x p = φ Dnr - φ re + ω a t, the ntegrated Doppler shft due to translaton of the center of the turntable relatve to the GPS satellte mnus the carrer NCO's 4
approxmate ntegrated Doppler shft plus a term that arses due to carrer phase wrap-up. Wrap-up s a combned effect of the sgnal's polarzaton and the antenna's atttude rotaton about ts feld-of-vew centerlne. The state x v = φ & Dnr + ω a, the Doppler shft due to the velocty of the turntable center relatve to the GPS satellte plus another carrer phase wrap-up term. The state x a = φ & Dnr, the rate of change of Doppler shft due to the acceleraton of the turntable center relatve to the GPS satellte. y RF (t) RF Front End y IF (t) sn[ ω IF t + φre ( t)] ω IF Carrer NCO y I (t) y Q (t) DLL w/carrer Adng cos[ ω IF t + φre ( t)] PLL Loop Flter Fg.. Block dagram of a sngle channel of a coherent GPS recever. The 3 vector w n- n eq. (6a) s the dscrete-tme whte nose process dsturbance. It models the effects of recever vehcle maneuvers. It has the followng statstcal model: E{w n- } = (7a) qa T E{ w m w n } = δ mn t n- qcs (7b) qcs where δ mn s the Kronecker delta and q a and q cs are equvalent contnuous-tme whte nose ntenstes. The modeled values of q a and q cs can be used to tune the resultng Kalman flter. The model n eqs. (6a) and (6b) allows Kalman flter estmaton of both the atttude parameters x c and x s and the velocty and acceleraton of the lne-of-sght from the center of the turntable to the tracked GPS satellte. The ablty to deal wth non-zero velocty and acceleraton s obvously mportant for vehcles. The ablty s also mportant for statc recevers because the moton of the GPS satellte causes sgnfcant relatve velocty and acceleraton. The measurement that s used n the Kalman flter s derved from the DLL's n-phase and quadrature accumulatons. It s a carrer phase error measurement: ω re I n Q n y n = - arctan(q n, I n ) (8) If the recever has acheved lock on the sgnal, then ths measurement can be modeled as the average dfference between the NCO's phase and the actual carrer phase. The average s taken over the tme nterval from t n- to t n : t n y n = [ x p( t) + xc ( t) cos( ωat + ψ a ) tn tn + xs ( t) sn( ω a t + ψ a )] dt + v n (9) where v n s a Gaussan random measurement error that s caused by thermal nose and dgtzaton. Its mean s zero, ts varance s σ v, and t s uncorrelated n tme and uncorrelated wth w n-. Ths measurement can be modeled n terms of the state vector of eq. (6a). The followng measurement equaton has been derved by substtuton nto eq. (9) of the underlyng contnuous-tme model that has been used to derve eq. (6a): y n = x p x t n t v n Cc( n ) Cs( n ) xa 6 xc x s n t - n ω re(n-) t + n Dc( n ) Ds( n ) w (n-) + v n () 4 The coeffcents n eq. () are sn( ω atn + ψ a) sn( ωatn + ψ a) C c(n-) = (a) ωa tn cos( ω atn+ ψ a) cos( ω atn + ψ a ) C s(n-) = - (b) ω a tn Cs( n ) sn(ω atn + ψ a) D c(n-) = - (c) ω a tn Cc( n ) cos(ω atn + ψ a) D s(n-) = (d) ωa tn The dscrete-tme model n eqs. (6a) and () takes the followng form: x n = Φ n- x n- + Γ n- ω re(n-) + Γ w(n-) w n- (a) y n = C n- x n- + D n- ω re(n-) + D w(n-) w n- + v n (b) The 5 state vector n ths model s x = [x p, x v, x a, x c, x s ] T. The matrces Φ n-, Γ n-, and Γ w(n-) are effectvely defned by eq. (6a), and the matrces C n-, D n-, and D w(n-) are defned by eq. (). 5
Ths tme-varyng system's 5 5 observablty Graman matrx has been calculated for one turntable rotaton perod. It has a rank of 5, whch proves the system's observablty 4. Therefore, the system states can be estmated from the carrer phase error measurements. Kalman Flter to Estmate Carrer Phase States A Kalman flter can be used to estmate the states of the phase model n eqs. (a) and (b). The Kalman flter keeps track of the estmated state vector, xˆ. It can be mplemented va the followng combned propagaton and update equatons: v n = y n - [C n- xˆ n + D n- ω re(n-) ] (3a) xˆ n = Φ n- xˆ n + Γ n- ω re(n-) + L n v n (3b) In these equatons the scalar v n s the flter nnovaton, and L n s the 5 flter gan matrx. The flter gan matrx wll be tme-varyng due to the tme varatons n the system model. The most mportant tme varatons are the snusodal varatons of C c(n-) and C s(n-), whch are elements of the C n- matrx. Normally L n would be computed usng a tme propagaton of a matrx Rccat equaton, whch, n ths case, would have to be specally desgned to account for the appearance of the process nose w n- n the measurement equaton 4. If the turntable's rotaton rate ω a s very slow or f the Kalman flter needs to have a hgh bandwdth, then ths way of computng L n wll defntely be needed. In the present case t s sometmes possble to compute a tme varyng flter gan wthout propagatng a matrx Rccat equaton. The followng flter gan s approxmately optmal when the Kalman flter's bandwdth s lower than the turntable rotaton speed: Lpva L n = LcsCc( n ) (4) LcsCs( n ) The quantty L pva s a constant 3 steady-state gan matrx. It can be derved by solvng a steady-state, tme-nvarant Kalman flter problem. Ths problem s for a modfed form of eqs. (6a) and (), one that deletes the states x c and x s and the nd and 3 rd elements of the process nose dsturbance vector w. Also, t n- s set equal to ts nomnal value of. sec. The scalar gan L cs can be determned by an averagng technque that solves a tme-nvarant Kalman flter problem for an average of the system over one perod of the turntable's rotaton, π/ω a. Such technques have been found to work well for perodc systems whose closed-loop bandwdth s low compared to the frequency of perodcty 5. Note that, n the case of a Kalman flter, loop closure refers to the feedng back of the nnovaton to correct the state estmate, as n eq. (3b). L cs s the scalar gan that would be used n the eq.-(3b) form of the steady-state Kalman flter for the followng scalar, tme-nvarant problem: x cs(n) = x cs(n-) + w cs(n-) (5a) y cs(n) =.5x cs(n-) +.5w cs(n-) + v cs(n) / (5b) where E{w cs(n-) } =, E{w cs(m-) w cs(n-) } = δ mn q cs (. sec), E{v cs(n) } =, E{v cs(m) v cs(n) } = δ mn σ v, and E{w cs(m- )v cs(n) } =. In ths model w cs(n-) s equvalent to the nd element of w n- n eq. (6a) and v cs(n) s equvalent to v n n eq. (). Equaton (5a) s equvalent to the fourth lne of eq. (6a), and eq. (5b) s equvalent to eq. () multpled by C c(n-) and averaged over a turntable rotaton perod. Alternatvely, w cs(n-) s equvalent to the 3 rd element of w n- n eq. (6a), and eqs. (5a) and (5b) can be derved by usng the ffth lne of eq. (6a) and by multplyng eq. () by C s(n-) and averagng over a turntable perod. Ths technque works because the flter s error dynamcs converge slowly compared to the turntable rotaton perod and because c( n ) s( n ) average( C ) = average( C ) =.5 whle average(c c(n-) ) = average(c s(n-) ) = average(c c(n-) C s(n-) ) =. These facts combne to yeld flters for the three state components [x p, x v, x a ] T, x c, and x s that are approxmately decoupled, and the composte gan for these three flters s well approxmated by the form gven n eq. (4). Use of the Kalman Flter Output to Drve the Carrer NCO The phase locked loop needs to feed back y, the phase error, to ω re, the frequency of the carrer trackng NCO. The Kalman flter, although t gves optmal estmates of the components of the phase error, gves no gudance on how to pck ω re. In theory, the flter can functon properly wth an arbtrary ω re. In practce, t s necessary to choose ω re so as to stablze y to a value near zero. Otherwse, cycle slps can occur due to the π ndetermnacy of eq. (8), or the assumptons of ths whole analyss can break down due to poor PRN code correlaton when the carrer NCO frequency s far dfferent from the ncomng sgnal s ntermedate frequency. The phase-locked loop s feedback control law uses the states of the Kalman flter to determne ω re. The PLL assumes that ω re(n) has already been chosen by the tme the Kalman flter s estmate xˆ s avalable. Therefore, t uses xˆ n to determne ω re(n+) accordng the followng rule: the predcted value of the phase error at tme t n+ must equal α tmes the estmated phase error at tme t n, where α s an arbtrary tunng factor for the n 6
PLL that s n the range α <. Ths rule s emboded n the followng formula for the NCO frequency: ω re(n+) = {- t n ω re(n) + (-α) xˆ p( n ) + ( t n + t n+ ) xˆ v( n ) +.5( t n + t n+ ) xˆ a( n ) + [cos(ω a t n+ +ψ a ) - α cos(ω a t n +ψ a )] xˆ c( n ) + [sn(ω a t n+ +ψ a ) - α sn(ω a t n +ψ a )] xˆ s( n ) }/ t n+ (6) The Kalman flter and ths ω re feedback law consttute the PLL loop flter that s shown n Fg.. The equatons that get mplemented n ths dgtal flter are eqs. (8), (3a), (3b), and (6). Equaton (8) can be mplemented effcently and wthout sgnfcant loss of SNR by usng an approxmaton to the -argument arctangent functon. PLL Tunng The feedback control law n eq. (6) wll cause the PLL to converge to zero phase error wth a frst-order response. The tme constant of ths decay wll be τ pll = -(. sec)/ln(α). The performance of the Kalman flter s theoretcally ndependent of the actual value of ths tme constant. τ pll should be chosen small enough to keep the phase errors from becomng too large, but not too small, otherwse system nose wll cause excessve jtter of the carrer NCO frequency. A value of α =.8 has been used throughout most of ths study, whch translates nto a settlng tme constant of τ pll = 9 msec. The overall bandwdth of the PLL s governed by τ pll and by the error decay tme constants of the Kalman flter. These latter tme constants are determned by the Kalman flter gan parameters L pva and L cs. When τ pll s small, the effectve bandwdth of the PLL s governed prmarly by the values of the Kalman flter gans. Dealng wth GPS Data Bts An actual system must be able to deal wth phase shfts that occur due to the transmsson of data bts. In a real GPS system, data bts are encoded on the sgnal at a rate of 5 bts/sec. Ths ntroduces the possblty of 8 deg phase shfts of φ Dopp once every PRN code perods. Extra logc s needed n a real recever to avod the possblty that the PLL wll nterpret such a phase shft as a change of φ Dopp due to an actual Doppler shft. The necessary logc s not hard to mplement f the recever s already trackng a sgnal that has a large SNR. The problem becomes trcker when one s tryng to acheve phase lock on a sgnal that has a low SNR. The problem of data bt logc s not addressed n the present paper. V. Smulaton of the GPS Sgnal and the Rotatng Antenna/Recever System A smulaton of ths system has been developed. It s for use n evaluatng the system s functonng and accuracy. The smulaton ncludes the followng components: the PRN-code-modulated sgnals of the tracked GPS satellte and of nterferng satelltes, the thermal and dgtzaton nose of the recever, the recever clock drft, the down-convertng mxers and band-pass flters of the RF front end, the carrer NCO, mxers, and loop flter of the PLL, and the PRN code NCO, correlators, and loop flter of the DLL. The smulaton mplements tme-doman models of the system s major elements. Thermal and dgtzaton nose typcally arse from dfferent elements wthn a crcut, but the smulaton lumps all of the nose at the recever nput and characterzes t by an equvalent total nput nose temperature. The recever nose temperature n the smulaton has been szed to match what has been observed expermentally n a terrestral applcaton. Ths experment used a typcal recever 8, a patch antenna wth a hemsphercal gan pattern, and a lownose preamplfer. The system's front end had a gan of 3 db and a nose fgure of.5 db as measured from the antenna nput to the recever nput. Most of the RF sgnals n the smulaton are represented by ther complex envelopes. A complex envelop representaton takes the form: j ω t y z (t) = real{ s z( t) e } (7) where y z (t) s a band-lmted sgnal n a frequency band centered at the carrer frequency ω, s z (t) s the baseband complex envelop of y z (t), and j =. It can be shown that any band-lmted sgnal can be represented n ths way, and t s straghtforward to model the effects of mxers and band-pass flters on a sgnal's complex envelop representaton 6. GPS Sgnal Model. The smulaton starts by constructng a complex envelop representaton of the ncomng GPS sgnal. Each GPS satellte sgnal conssts of a sne wave that s Doppler shfted from the nomnal 575.4 MHz L carrer frequency and that has ts pseudo-random code modulated onto t va bnary phase-shft keyng. Suppose that the ncomng sgnal s y RF (t) and that ts complex envelop s s RF (t), smlar to eq. (7). Then the center frequency s ω = ω c = π 575.4 6 rad/sec, and the complex envelop s N s RF (t) = { A C [ ( t)] exp[ φ ( t)] } = τ j + v RF (t) (8) Dopp The superscrpt n ths equaton refers to GPS 7
satellte. The ampltude A sets the sgnal power. The functon C (τ ) s the satellte's ± pseudorandom code, and τ (t) s the pseudo-random code phase measured n code seconds. The PRN code repeats tself wth a perod of τ =. code sec. Dopp The quantty φ (t) s the ntegrated Doppler shft of the carrer sgnal. Note that τ (t) = t + [ φ Dopp(t) /ω c ] + constant. The sgnal v RF (t) s a complex envelop representaton of the equvalent total thermal and dgtzaton nose of the antenna plus the recever. The smulaton only attempts to track one of the GPS satellte sgnals. The others are ncluded to smulate mult-channel nterference. Each φ Dopp(t) tme hstory s determned from the GPS satellte range tme hstory and the table rotaton tme hstory accordng to the followng formulas, whch are consstent wth eqs. () and (): φ Dopp(t) = constant - (ω c /c) [ ρ s( t)] + ra ρ s( t) ra cosθs cos( ω a t + ψ a ψ s ) (9) where ρ s (t) = ρ s + ρ& s t +.5 ρ& & s t () s ρ s the ntal dstance from the turntable center to GPS satellte, ρ& s s the ntal range rate, and ρ& & s s the range acceleraton. The smulaton assumes that each GPS satellte has a nonzero ntal lne-of-sght velocty, ρ& s, and a nonzero lne-of-sght acceleraton, ρ& & s. The ntal lne- of-sght rates have been chosen randomly to fall n the range ± 8,6 m/sec, whch s consstent wth the possble range of relatve veloctes for a user satellte n low Earth orbt. The lne-of-sght acceleratons have been chosen randomly to fall n the range ±5 g's. Although ths large range for the acceleratons s probably excessve, t serves to make the pont that large acceleratons do not adversely affect the system's performance. The smulaton uses the actual GPS C/A pseudorandom codes. They are generated by computer code that emulates smple feedback shft regsters 7. The smulaton uses a sampled verson of the sgnal. If the sample nterval s defned to be t sm, then the sampled sgnal s s RF(m) = s RF (m t sm ) N = { A C [ τ ( m t )] exp[ jφ ( m t )]} = sm Dopp sm + v RF(m) () where m s the sample ndex and v RF(m) s a sampled-data verson of the RF nose model. The nomnal sample perod that has been used s t sm = 8.46 nanosec. Ths yelds samples per pseudo-random code chp, whch s adequate to represent the dgtal code sgnals C (τ ). The samplng frequency s / t sm =.3 MHz. Ths s sgnfcantly more than twce the MHz bandwdth of the ntermedate RF sgnal that comes out of each recever s RF front end, y IF (t), whch mples that ths sample perod s adequately small. The thermal/dgtzaton nose model s represented by the dscrete-tme Gaussan whte-nose sequence v RF(), v RF(), v RF(),..., v RF(m), Its standard devaton s σ nose = k T / t () rcvr sm where k s Boltzmann's constant and T rcvr s the equvalent nput nose temperature of the recever n degrees Kelvn. The dscrete-tme nose sequence s then n m = σ nose [ν real(m) + jν mag(m) ] (3) In ths model ν real(m) and ν mag(m) are both real, zero-mean, unt-varance, uncorrelated dscrete-tme whte-nose processes, whch are smulated by a random number generator. One sgnfcant error source has been neglected n ths smulaton model, mult-path nose. It has been neglected because t s too dffcult to model. It s best studed va experment. For completeness sake, however, a later secton of ths paper estmates the magntude of mult-path-nduced atttude measurement errors. Complex Envelop Smulaton of the Recever s RF Front End. The RF front end of the recever s modeled by 3 stages of band-pass flterng that alternate wth two stages of mxng. It produces a sgnal y IF (t) whose ntermedate frequency s nomnally 3.636 MHz and whose bandwdth s approxmately MHz. The mxers and flters are modeled by approprate band-pass modelng technques 6. The last band-pass flter model, the one wth a MHz bandwdth, has a 5-pole complex envelop representaton. Its 5 poles fall n an asymmetrcal Butterworth-lke pattern. Its model matches expermental frequency response data from an actual flter. PLL and DLL Smulaton. Ths secton descrbes the smulaton of everythng that s downstream of the RF front-end: the n-phase and quadrature mxers, the DLL, the PLL's loop flter, and the carrer NCO. The nput to ths part of the smulaton s y IF (t), the ntermedate frequency sgnal that comes out of the 8
recever s RF front end revew Fg.. Ths part of the smulaton bases ts calculatons on tme ntervals, each of whch corresponds to the recever s estmate of a dstnct perod of the PRN code of the tracked satellte. The boundares of these tme ntervals are t, t, t,, t n, At these sample tmes the recever s estmated code phase s always an nteger multple of the nomnal code perod of. sec.;.e., ˆτ ( t n ) = n., where ˆτ (t) s the DLL s estmated code phase at tme t. Ths part of the smulaton works entrely wth real sgnals. Before mxng the sgnal to base-band, t uses the complex envelop s IF (t) to compute the actual sgnal that comes out of the RF front-end: y IF (t) = j ω t real{ s IF IF ( t) e }. The next operaton s the generaton of the outputs of the n-phase and quadrature carrer mxers. They are y I (t) = cos[ω IF t + ω re(n-) (t t n- ) + φ re(n-) ] y IF (t) for t n- t < t n y Q (t) = - sn[ω IF t + ω re(n-) (t t n- ) + φ re(n-) ] y IF (t) (4a) for t n- t < t n (4b) The value ω IF = π 3.636 6 rad/sec s used as the nomnal mxng frequency. The next part of the smulaton models the PRN code play-back NCO, the code mxers, and the ntegrate-and-dump accumulators. As far as the PLL s concerned, these actons effectvely perform the followng calculatons to determne the n-phase and quadrature accumulatons: t n d ˆ I (n) = y t C ˆ τ I ( ) [ τ ( t)] dt. dt (5a) Q (n) = tn t n yq tn. d ˆ ˆ τ ( t) C[ τ ( t)] dt dt (5b) where the PRN code, C[τ], and the estmated code phase, ˆτ (t), both correspond to the tracked GPS satellte. The actual calculatons are dgtal summatons that approxmate these ntegrals. The summatons break the ntegraton ntervals up nto 3 sub-ntervals and perform Euler ntegraton. The smulaton also emulates a DLL. Ths nvolves the calculaton of early and late accumulatons, smlar to eqs. (5a) and (5b) but wth offsets of τˆ. Another part of the DLL smulaton s a carrer-aded proportonal feedback control law that adjusts dτˆ /dt of ts PRN play-back NCO. The goal of ths feedback controller s to algn ˆτ (t) wth the actual τ(t) of the receved sgnal's code. The control law s ω ˆ n +.5 = xˆ v( n ) + (.5 tn ) xˆ a( n ) ω xˆ sn[ ω ( t +.5 t ) + ψ ] + a { c( n ) a n n a + xˆ s( n ) cos[ ω a( tn +.5 tn) + ψ a]} (6a) dτˆ ω ˆ dt = + K DLL (τ-τˆ ) n + n +.5 (6b) n + ω c K DLL s the proportonal gan. A value of π has been used for K DLL, whch corresponds to a Hz bandwdth for the DLL. The code phase error term, (τ-τˆ ) n, s computed from the dfference between early and late accumulatons for the tme nterval t n- to t n. Ths type of computaton s descrbed n Ref.. The last term on the rght-hand sde of eq. (6b) s the carrer adng term; ω ˆ n +. 5 s a predcton of what the sgnal's average Doppler shft wll be durng the tme nterval from t n+ to t n+. The smulaton's PLL loop flter calculatons have already been descrbed n Secton IV. They are gven n eqs. (8), (3a), (3b), and (6). The effects of recever clock errors have been ncorporated usng a two-state drft model from Ref. 8. The states are δt rc, the tme error, and δf rc, the fractonal frequency error. Ther dynamc models are: δ t& rc = δf rc + w rc and δ f & rc = w rc. The whte nose processes w rc and w rc drve the drft wth ntenstes E{w rc (t)w rc (τ)} =.5h δ(t-τ) and E{w rc (t)w rc (τ)} = π h - δ(t-τ). The constants h and h - defne the level of drft. Ths clock drft model mpacts the rest of the smulaton through eq. (8). The average value of the quantty ω c δt rc durng the accumulaton nterval from t n- to t n gets subtracted from the rght-hand sde of eq. (8). Ths models the effect of recever clock drft on the PLL's measurement of the carrer phase error. The smulaton operates teratvely and must be ntalzed. It processes one PRN code perod nterval at a tme, and t uses some of the outputs from the nterval t n- to t n as the nputs to the nterval t n to t n+. In order to ntalze the process, the smulaton needs to start wth guesses of the Kalman flter state, xˆ, the NCO phase, φ re, the estmated PRN code phase, τˆ, the PLL's NCO rate, ω re, and the DLL's NCO rate, dτˆ /dt. In order to acheve lock, these guesses need to be farly accurate. Any real recever has a start-up mode that searches to fnd good ntal guesses for such quanttes. After the search s complete, the recever locks onto the GPS sgnal and tracks t for a long tme. The system's acquston-mode performance does not mpact ts accuracy durng ths steady-state perod. Therefore, farly good frst guesses of the above quanttes have been used, and the ssue of sgnal acquston has not been consdered n the present study. 9
VI. Evaluaton of Atttude Sensng Performance The smulaton and mscellaneous analyses have been used to evaluate the atttude sensng accuracy of the system. There are a number of ssues that have been nvestgated n order to determne the system's expected performance. Thermal and dgtzaton nose, nterference from other GPS satelltes, recever clock drft, and recever dstorton all mght cause atttude sensng errors. The amount by whch these effects degrade accuracy needs to be nvestgated. Some of these errors can be reduced by reducng the atttude sensng bandwdth, and the relatonshp between bandwdth and accuracy must be determned. Other system parameters that may affect accuracy are turntable rotaton speed, ω a, antenna mountng radus, r a, and the elevaton angle of the tracked GPS satellte above the turntable's plane of rotaton, θ s. These parameters' effects also need to be nvestgated. Also at ssue s whether the system can dstngush small perodc ntegrated Doppler shft varatons that rde on top of the Doppler shfts that are caused by large lneof-sght veloctes and acceleratons. An example case has been evaluated usng the smulaton. It s characterzed by the followng parameters: The antenna mountng radus s r a =. m, and the turntable speed s ω a = 49 rad/sec (4 RPM). The tracked GPS satellte has an elevaton angle of θ s = π/4 rad (45 deg). The thermal/dgtzaton nose level of the antenna and recever combne wth the level of the receved sgnal's power to yeld an SNR of 8 db for the I n and Q n accumulator outputs. Ths SNR level corresponds to a 7.-deg RMS phase measurement nose at the Hz samplng frequency. The clock drft parameters are those of a representatve ovenzed crystal oscllator: h = - sec and h - = 6. - /sec. The gan for the atttude sensng part of the Kalman flter s L cs =.8, whch yelds a.64 Hz atttude sensng bandwdth. The flter gan component L pva equals [.46,.935, 9.7869] T. Ths produces a 3.4 Hz velocty/acceleraton determnaton bandwdth, and the characterstc values of ths part of the Kalman flter form a 3-pole Butterworth pattern. The ntal velocty and acceleraton of the lne of sght from the turntable center to the tracked GPS satellte are ρ& s = 6 m/sec and ρ& & s = 5 m/sec (5 g's). The tracked satellte transmts PRN code number 8. There are 8 nterferng GPS satelltes, all wth the same receved power level as the tracked satellte. These nterferng sgnals reduce the SNR at the I n and Q n outputs by db. Fgure 3 shows the estmaton errors for θ s (sold lne) and ψ s (dash-dotted lne) for ths case. Durng the frst half second of the smulaton the Kalman flter converges from ntal errors. Afterwards, t settles nto a steady state. Both angles have steady-state RMS errors of. deg and peak steady-state errors of about.6 deg. Ths s relatvely coarse atttude accuracy. Atttud Error (deg) Fg. 3. Atttude estmaton errors for a typcal case. In order to explore system accuracy n varous stuatons, t s helpful to parameterze the atttude n terms of the unt drecton vector to the tracked GPS spacecraft, rˆ s. One can use x c and x s to drectly calculate ths vector. Equatons (4a) and (4b) and the geometry of Fg. mply that c x ω c c ra c xs (7) ωc ra ( ) + c xc xs ωcra The covarance of the rˆ s estmaton error can be rˆ s =.5 -.5 Elevaton Error Azmuth Error - 4 6 8 Tme (sec) calculated from the covarance of the x c and x s estmaton errors. Suppose that these latter two quanttes' estmaton errors are uncorrelated and that ther standard devatons both equal σ cs. (Ths s a good approxmaton for the Kalman flter whose gan s gven n eq. (4) f the turntable rotaton speed s hgh enough.) Then the rˆ s vector's (lnearzed) estmaton error covarance matrx s E{[ rˆ s( est ) - rˆ s( actual) ][ rˆ s( est ) - rˆ s( actual) ] T } = σ cs c T T rˆ + r ψ rˆ ψ rˆ rˆ θ θ (8) ωc a sn θs where rˆ ψ = [-snψ s, cosψ s, ] T and rˆ θ = [-cosψ s snθ s, - snψ s snθ s, cosθ s ] T are orthogonal unt vectors n the
drectons of locally ncreasng ψ s and θ s, respectvely. Equaton (8) gves the key to understandng the effects on the system's accuracy of varous desgn parameters. The drectonal standard devatons for rˆ s estmaton errors are: σ cs c σ el = (9a) ωcra snθ s σ az = σ csc (9b) ωcra where σ el s measured n the elevaton drecton and σ az s measured n the azmuthal drecton. The smulatons and analyses have shown that σ cs, the standard devaton of x c and x s, depends only on the flter's bandwdth and the nput sgnal-to-nose rato for the channel. The quanttes c and ω c are fxed parameters. Therefore, the only ways to affect system accuracy are to change the antenna mountng radus on the turntable, r a, the bandwdth of the flter, or the nput sgnal-to-nose rato of the recever/antenna system. Equaton (9a) says that the elevaton accuracy ncreases wth ncreasng elevaton, reachng ts maxmum when θ s = 9 deg. At θ s = deg the elevaton standard devaton goes to nfnty. In other words, the best vector atttude sensng geometry occurs when the tracked GPS satellte s lned up on the turntable's rotaton axs, and the worst geometry occurs when the tracked satellte s n the turntable plane revew Fg.. There s obvously a lower bound on the useable elevaton wndow f one wants to get a reasonable vector atttude measurement from the system. Elevatons down as low as θ s = 45 deg are certanly useable, as demonstrated by the results n Fg. 3. At 45 deg, the RMS elevaton error s only 4% larger than t s at θ s = 9 deg. A full 3-axs atttude soluton requres that the system track or more non-collnear GPS satelltes. Therefore, at least some of the tracked GPS satelltes must have elevatons sgnfcantly below 9 deg. Ths requrement should not present a problem. In fact, t would be acceptable to track a satellte wth very low turntable-relatve elevaton f azmuth nformaton was the only nformaton whch was needed from that satellte. The smulaton results have borne out ths analyss. When r a s ncreased, atttude errors decrease proportonately. When θ s s ncreased, elevaton errors decrease as /snθ s. The use of dfferent turntable speeds does not affect atttude error, to a certan pont, f one uses an ovenzed crystal oscllator wth stablty characterstcs lke those that have been used n the Fg.-3 example. Turntable speeds as low as 5 rad/sec. ( RPM) have been smulated wthout changng any other parameters, and the system's performance has been vrtually unchanged. Of course, f very low turntable speeds are used, then the atttude error wll be affected, ether because of excessve recever clock drft or because of volaton of the bandwdth assumpton that s assocated wth the perodc Kalman flter gan approxmaton n eq. (4). The effect of recever clock stablty has been nvestgated. The example that produced Fg. 3 has been re-run wth a less stable recever clock, one whose drft model s characterzed by the parameters h = -9 sec and h - = - /sec. Ths corresponds to a temperature-compensated crystal oscllator 8. The atttude determnaton performance deterorates sgnfcantly n ths case. The RMS elevaton and azmuth errors ncrease to.5 deg. Furthermore, the system senstvty to rotor speed ncreases when ths poorer recever clock s used. If the wheel speed s decreased from 4, RPM to, RPM, then the RMS atttude errors ncrease to about.5 deg. It should be noted at ths pont that there s a way to estmate and compensate for the effects of recever clock errors. The method reles on atttude data from 3 or more GPS satelltes that are n a sutable geometrc relatonshp. The recever-clock-nduced errors n xˆ c and xˆ s are the same for all tracked satelltes. They can be estmated by comparng the measured angles between the vectors to the tracked GPS satelltes wth the known values for these angles based on ephemerdes. Ths technque s analogous to the standard recever clock correcton that s used n the GPS navgaton soluton. The full analyss of ths technque s not gven here because t s beyond the scope of ths paper. The smulaton results show that performance s nsenstve to several envronmental and system dsturbances. Interference from other GPS satelltes does not sgnfcantly affect performance at practcally achevable SNRs. Recever dstorton n the RF front end also has lttle effect. The atttude sensng accuracy does not degrade f large Doppler shfts and Doppler shft rates get nduced by large veloctes and acceleratons of the turntable s center. Atttude sensng accuracy can be ncreased by lowerng the flter bandwdth or by rasng the SNR. A smple calculaton shows that σ cs scales as the square root of flter bandwdth dvded by sgnal-to-nose rato, and the smulaton results have borne ths out. Suppose that one wanted to mprove the accuracy of the case assocated wth Fg. 3. Suppose that the accuracy goal was to reduce the RMS elevaton error
to.3 deg. Recall that Fg. 3 shows a steady-state RMS elevaton error of. deg and that the atttude sensng flter bandwdth s.64 Hz. The flter bandwdth would have to be reduced to.4 Hz n order to meet the.3 deg accuracy goal. Such a flter would have an effectve delay of sec when operatng on a dynamcally varyng atttude sgnal, whch would be unacceptable performance n many stuatons. The flter assocated wth Fg. 3 has a delay of only.5 sec. If one wanted to make the same accuracy mprovement va reducton of recever nose, then an ncrease of 6 db n the SNR would be needed. Such an ncrease s probably not feasble. Another way to ncrease accuracy s to lower the flter bandwdth whle augmentng the atttude determnaton system wth an nertal atttude measurement. One could add a tunng-fork rate gyro n order to get acceptable bandwdth wth respect to real dynamc atttude varatons whle smultaneously lowerng the bandwdth of the GPS part of the system. Ths type of approach has been tred successfully wth a mult-antenna-based GPS atttude sensng system 9, and t would probably work well wth the rotatngantenna system. Such an approach would work best f the atttude determnaton Kalman flter were coupled to the recever s PLL. There may be a practcal way to ncrease accuracy by drastcally ncreasng the mountng radus of the antenna, r a. There are practcal lmtatons to the sze of a physcal turntable, but these lmtatons can be overcome f one electrcally smulates an antenna on a turntable. One way to do ths would be to mount a large rng of patch antennas on the user vehcle. An RF swtchng crcut would connect them to the recever one at a tme. The sequence of connecton would follow a crcular path, whch would synthesze crcular moton of the phase center of a sngle antenna. Ths s one of the methods descrbed n Ref. 7 for rado drecton fndng. Note that t should be possble to use dstorted crcular patterns. Such a mountng pattern mght be more easly realzable due to antenna locaton constrants on a real vehcle. If the mountng pattern s not exactly crcular, then the recever's PLL wll have to be modfed to account for the dstorton. One mght protest that such a desgn would be a reverson to the orgnal mult-antenna approach. On one level ths s true, but such a system would retan the advantage of not needng many recever channels. Furthermore, f the neghborng antennas were close enough to each other, then the system would not need to resolve nteger ambgutes, yet t would have the advantages of a long baselne. Of course, ths approach assumes that t s practcal to mount many patch antennas and a number of RF swtches on the user vehcle. It should be noted that, n some sense, ths paper's new system s equvalent to the orgnal mult-antenna GPS atttude sensng scheme. Instead of usng multple antennas, t uses one antenna at multple locatons. Although ths scheme has several advantages, t retans some of the basc lmtatons of the mult-antenna system. The most mportant common lmtaton s that both systems accuraces vary n the same way wth bandwdth and wth antenna baselne length (r a ). Ths analogy allows one to make a rough estmate of the mpact of mult-path errors on accuracy. Multpath has been found to nduce.5 m RMS dfferental carrer phase rangng errors between pars of statc antennas. It s reasonable to suppose that ths dfferental error magntude wll hold true for the present system's sngle antenna f the dfference s taken between tmes when the antenna s on opposte sdes of ts crcular path. In ths case, the atttude error wll be.5m/(r a ) radans. For r a =. m ths translates nto an RMS atttude error of.4 deg. Thus, mult-path error wll domnate all other error sources unless the system uses a low-stablty oscllator n ts recever and a low turntable speed. Mult-path errors can be reduced by ncreasng, r a, the mountng radus of the antenna. Conclusons A system that senses vector atttude nformaton usng a sngle GPS antenna has been proposed and analyzed. The antenna s mounted on a turntable wth ts phase center offset from the rotaton axs. The resultng crcular moton causes perodc phase modulaton of the receved GPS carrer sgnal. Ths perodc modulaton can be detected by usng a specal loop flter n the recever s carrer trackng phaselocked loop. The ampltude and phase of the modulaton can be used to deduce the drecton vector to the tracked GPS satellte n recever vehcle coordnates. Ths vector measurement s an atttude measurement, and two such vector measurements are suffcent to determne the 3-axs atttude of the user vehcle. The proposed system has been analyzed, and t has been evaluated usng a tme-doman smulaton. A -σ accuracy of.4 deg has been demonstrated for a system that uses a. m antenna mountng radus, a 4, RPM turntable speed, an ovenzed crystal oscllator for ts recever clock, and a.64 Hz phaselocked loop bandwdth. The atttude error standard devaton s nversely proportonal to the antenna mountng radus. The accuracy s essentally ndependent of the turntable rotaton speed f that